text: """ Test functions for stats module
  
      WRITTEN BY LOUIS LUANGKESORN <lluang@yahoo.com> FOR THE STATS MODULE
      BASED ON WILKINSON'S STATISTICS QUIZ
      https://www.stanford.edu/~clint/bench/wilk.txt
  
      Additional tests by a host of SciPy developers.
  """
  import os
  import re
  import warnings
  from collections import namedtuple
  from itertools import product
  
  from numpy.testing import (assert_, assert_equal,
                             assert_almost_equal, assert_array_almost_equal,
                             assert_array_equal, assert_approx_equal,
                             assert_allclose, assert_warns, suppress_warnings,
                             assert_array_less)
  import pytest
  from pytest import raises as assert_raises
  import numpy.ma.testutils as mat
  from numpy import array, arange, float32, float64, power
  import numpy as np
  
  import scipy.stats as stats
  import scipy.stats.mstats as mstats
  import scipy.stats._mstats_basic as mstats_basic
  from scipy.stats._ksstats import kolmogn
  from scipy.special._testutils import FuncData
  from scipy.special import binom
  from .common_tests import check_named_results
  from scipy.sparse._sputils import matrix
  from scipy.spatial.distance import cdist
  from numpy.lib import NumpyVersion
  from scipy.stats._axis_nan_policy import _broadcast_concatenate
  from scipy.stats._stats_py import _permutation_distribution_t
  
  
  """ Numbers in docstrings beginning with 'W' refer to the section numbers
      and headings found in the STATISTICS QUIZ of Leland Wilkinson.  These are
      considered to be essential functionality.  True testing and
      evaluation of a statistics package requires use of the
      NIST Statistical test data.  See McCoullough(1999) Assessing The Reliability
      of Statistical Software for a test methodology and its
      implementation in testing SAS, SPSS, and S-Plus
  """
  
  #  Datasets
  #  These data sets are from the nasty.dat sets used by Wilkinson
  #  For completeness, I should write the relevant tests and count them as failures
  #  Somewhat acceptable, since this is still beta software.  It would count as a
  #  good target for 1.0 status
  X = array([1,2,3,4,5,6,7,8,9], float)
  ZERO = array([0,0,0,0,0,0,0,0,0], float)
  BIG = array([99999991,99999992,99999993,99999994,99999995,99999996,99999997,
               99999998,99999999], float)
  LITTLE = array([0.99999991,0.99999992,0.99999993,0.99999994,0.99999995,0.99999996,
                  0.99999997,0.99999998,0.99999999], float)
  HUGE = array([1e+12,2e+12,3e+12,4e+12,5e+12,6e+12,7e+12,8e+12,9e+12], float)
  TINY = array([1e-12,2e-12,3e-12,4e-12,5e-12,6e-12,7e-12,8e-12,9e-12], float)
  ROUND = array([0.5,1.5,2.5,3.5,4.5,5.5,6.5,7.5,8.5], float)
  
  
  class TestTrimmedStats:
      # TODO: write these tests to handle missing values properly
      dprec = np.finfo(np.float64).precision
  
      def test_tmean(self):
          y = stats.tmean(X, (2, 8), (True, True))
          assert_approx_equal(y, 5.0, significant=self.dprec)
  
          y1 = stats.tmean(X, limits=(2, 8), inclusive=(False, False))
          y2 = stats.tmean(X, limits=None)
          assert_approx_equal(y1, y2, significant=self.dprec)
  
          x_2d = arange(63, dtype=float64).reshape(9, 7)
          y = stats.tmean(x_2d, axis=None)
          assert_approx_equal(y, x_2d.mean(), significant=self.dprec)
  
          y = stats.tmean(x_2d, axis=0)
          assert_array_almost_equal(y, x_2d.mean(axis=0), decimal=8)
  
          y = stats.tmean(x_2d, axis=1)
          assert_array_almost_equal(y, x_2d.mean(axis=1), decimal=8)
  
          y = stats.tmean(x_2d, limits=(2, 61), axis=None)
          assert_approx_equal(y, 31.5, significant=self.dprec)
  
          y = stats.tmean(x_2d, limits=(2, 21), axis=0)
          y_true = [14, 11.5, 9, 10, 11, 12, 13]
          assert_array_almost_equal(y, y_true, decimal=8)
  
          y = stats.tmean(x_2d, limits=(2, 21), inclusive=(True, False), axis=0)
          y_true = [10.5, 11.5, 9, 10, 11, 12, 13]
          assert_array_almost_equal(y, y_true, decimal=8)
  
          x_2d_with_nan = np.array(x_2d)
          x_2d_with_nan[-1, -3:] = np.nan
          y = stats.tmean(x_2d_with_nan, limits=(1, 13), axis=0)
          y_true = [7, 4.5, 5.5, 6.5, np.nan, np.nan, np.nan]
          assert_array_almost_equal(y, y_true, decimal=8)
  
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "Mean of empty slice")
  
              y = stats.tmean(x_2d, limits=(2, 21), axis=1)
              y_true = [4, 10, 17, 21, np.nan, np.nan, np.nan, np.nan, np.nan]
              assert_array_almost_equal(y, y_true, decimal=8)
  
              y = stats.tmean(x_2d, limits=(2, 21),
                              inclusive=(False, True), axis=1)
              y_true = [4.5, 10, 17, 21, np.nan, np.nan, np.nan, np.nan, np.nan]
              assert_array_almost_equal(y, y_true, decimal=8)
  
      def test_tvar(self):
          y = stats.tvar(X, limits=(2, 8), inclusive=(True, True))
          assert_approx_equal(y, 4.6666666666666661, significant=self.dprec)
  
          y = stats.tvar(X, limits=None)
          assert_approx_equal(y, X.var(ddof=1), significant=self.dprec)
  
          x_2d = arange(63, dtype=float64).reshape((9, 7))
          y = stats.tvar(x_2d, axis=None)
          assert_approx_equal(y, x_2d.var(ddof=1), significant=self.dprec)
  
          y = stats.tvar(x_2d, axis=0)
          assert_array_almost_equal(y[0], np.full((1, 7), 367.50000000), decimal=8)
  
          y = stats.tvar(x_2d, axis=1)
          assert_array_almost_equal(y[0], np.full((1, 9), 4.66666667), decimal=8)
  
          y = stats.tvar(x_2d[3, :])
          assert_approx_equal(y, 4.666666666666667, significant=self.dprec)
  
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "Degrees of freedom <= 0 for slice.")
  
              # Limiting some values along one axis
              y = stats.tvar(x_2d, limits=(1, 5), axis=1, inclusive=(True, True))
              assert_approx_equal(y[0], 2.5, significant=self.dprec)
  
              # Limiting all values along one axis
              y = stats.tvar(x_2d, limits=(0, 6), axis=1, inclusive=(True, True))
              assert_approx_equal(y[0], 4.666666666666667, significant=self.dprec)
              assert_equal(y[1], np.nan)
  
      def test_tstd(self):
          y = stats.tstd(X, (2, 8), (True, True))
          assert_approx_equal(y, 2.1602468994692865, significant=self.dprec)
  
          y = stats.tstd(X, limits=None)
          assert_approx_equal(y, X.std(ddof=1), significant=self.dprec)
  
      def test_tmin(self):
          assert_equal(stats.tmin(4), 4)
  
          x = np.arange(10)
          assert_equal(stats.tmin(x), 0)
          assert_equal(stats.tmin(x, lowerlimit=0), 0)
          assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), 1)
  
          x = x.reshape((5, 2))
          assert_equal(stats.tmin(x, lowerlimit=0, inclusive=False), [2, 1])
          assert_equal(stats.tmin(x, axis=1), [0, 2, 4, 6, 8])
          assert_equal(stats.tmin(x, axis=None), 0)
  
          x = np.arange(10.)
          x[9] = np.nan
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "invalid value*")
              assert_equal(stats.tmin(x), np.nan)
              assert_equal(stats.tmin(x, nan_policy='omit'), 0.)
              assert_raises(ValueError, stats.tmin, x, nan_policy='raise')
              assert_raises(ValueError, stats.tmin, x, nan_policy='foobar')
              msg = "'propagate', 'raise', 'omit'"
              with assert_raises(ValueError, match=msg):
                  stats.tmin(x, nan_policy='foo')
  
      def test_tmax(self):
          assert_equal(stats.tmax(4), 4)
  
          x = np.arange(10)
          assert_equal(stats.tmax(x), 9)
          assert_equal(stats.tmax(x, upperlimit=9), 9)
          assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), 8)
  
          x = x.reshape((5, 2))
          assert_equal(stats.tmax(x, upperlimit=9, inclusive=False), [8, 7])
          assert_equal(stats.tmax(x, axis=1), [1, 3, 5, 7, 9])
          assert_equal(stats.tmax(x, axis=None), 9)
  
          x = np.arange(10.)
          x[6] = np.nan
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "invalid value*")
              assert_equal(stats.tmax(x), np.nan)
              assert_equal(stats.tmax(x, nan_policy='omit'), 9.)
              assert_raises(ValueError, stats.tmax, x, nan_policy='raise')
              assert_raises(ValueError, stats.tmax, x, nan_policy='foobar')
  
      def test_tsem(self):
          y = stats.tsem(X, limits=(3, 8), inclusive=(False, True))
          y_ref = np.array([4, 5, 6, 7, 8])
          assert_approx_equal(y, y_ref.std(ddof=1) / np.sqrt(y_ref.size),
                              significant=self.dprec)
  
          assert_approx_equal(stats.tsem(X, limits=[-1, 10]),
                              stats.tsem(X, limits=None),
                              significant=self.dprec)
  
  
  class TestCorrPearsonr:
      """ W.II.D. Compute a correlation matrix on all the variables.
  
          All the correlations, except for ZERO and MISS, should be exactly 1.
          ZERO and MISS should have undefined or missing correlations with the
          other variables.  The same should go for SPEARMAN correlations, if
          your program has them.
      """
  
      def test_pXX(self):
          y = stats.pearsonr(X,X)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pXBIG(self):
          y = stats.pearsonr(X,BIG)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pXLITTLE(self):
          y = stats.pearsonr(X,LITTLE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pXHUGE(self):
          y = stats.pearsonr(X,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pXTINY(self):
          y = stats.pearsonr(X,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pXROUND(self):
          y = stats.pearsonr(X,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pBIGBIG(self):
          y = stats.pearsonr(BIG,BIG)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pBIGLITTLE(self):
          y = stats.pearsonr(BIG,LITTLE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pBIGHUGE(self):
          y = stats.pearsonr(BIG,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pBIGTINY(self):
          y = stats.pearsonr(BIG,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pBIGROUND(self):
          y = stats.pearsonr(BIG,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pLITTLELITTLE(self):
          y = stats.pearsonr(LITTLE,LITTLE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pLITTLEHUGE(self):
          y = stats.pearsonr(LITTLE,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pLITTLETINY(self):
          y = stats.pearsonr(LITTLE,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pLITTLEROUND(self):
          y = stats.pearsonr(LITTLE,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pHUGEHUGE(self):
          y = stats.pearsonr(HUGE,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pHUGETINY(self):
          y = stats.pearsonr(HUGE,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pHUGEROUND(self):
          y = stats.pearsonr(HUGE,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pTINYTINY(self):
          y = stats.pearsonr(TINY,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pTINYROUND(self):
          y = stats.pearsonr(TINY,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_pROUNDROUND(self):
          y = stats.pearsonr(ROUND,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_r_almost_exactly_pos1(self):
          a = arange(3.0)
          r, prob = stats.pearsonr(a, a)
  
          assert_allclose(r, 1.0, atol=1e-15)
          # With n = len(a) = 3, the error in prob grows like the
          # square root of the error in r.
          assert_allclose(prob, 0.0, atol=np.sqrt(2*np.spacing(1.0)))
  
      def test_r_almost_exactly_neg1(self):
          a = arange(3.0)
          r, prob = stats.pearsonr(a, -a)
  
          assert_allclose(r, -1.0, atol=1e-15)
          # With n = len(a) = 3, the error in prob grows like the
          # square root of the error in r.
          assert_allclose(prob, 0.0, atol=np.sqrt(2*np.spacing(1.0)))
  
      def test_basic(self):
          # A basic test, with a correlation coefficient
          # that is not 1 or -1.
          a = array([-1, 0, 1])
          b = array([0, 0, 3])
          r, prob = stats.pearsonr(a, b)
          assert_approx_equal(r, np.sqrt(3)/2)
          assert_approx_equal(prob, 1/3)
  
      def test_constant_input(self):
          # Zero variance input
          # See https://github.com/scipy/scipy/issues/3728
          with assert_warns(stats.PearsonRConstantInputWarning):
              r, p = stats.pearsonr([0.667, 0.667, 0.667], [0.123, 0.456, 0.789])
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
  
      def test_near_constant_input(self):
          # Near constant input (but not constant):
          x = [2, 2, 2 + np.spacing(2)]
          y = [3, 3, 3 + 6*np.spacing(3)]
          with assert_warns(stats.PearsonRNearConstantInputWarning):
              # r and p are garbage, so don't bother checking them in this case.
              # (The exact value of r would be 1.)
              r, p = stats.pearsonr(x, y)
  
      def test_very_small_input_values(self):
          # Very small values in an input.  A naive implementation will
          # suffer from underflow.
          # See https://github.com/scipy/scipy/issues/9353
          x = [0.004434375, 0.004756007, 0.003911996, 0.0038005, 0.003409971]
          y = [2.48e-188, 7.41e-181, 4.09e-208, 2.08e-223, 2.66e-245]
          r, p = stats.pearsonr(x,y)
  
          # The expected values were computed using mpmath with 80 digits
          # of precision.
          assert_allclose(r, 0.7272930540750450)
          assert_allclose(p, 0.1637805429533202)
  
      def test_very_large_input_values(self):
          # Very large values in an input.  A naive implementation will
          # suffer from overflow.
          # See https://github.com/scipy/scipy/issues/8980
          x = 1e90*np.array([0, 0, 0, 1, 1, 1, 1])
          y = 1e90*np.arange(7)
  
          r, p = stats.pearsonr(x, y)
  
          # The expected values were computed using mpmath with 80 digits
          # of precision.
          assert_allclose(r, 0.8660254037844386)
          assert_allclose(p, 0.011724811003954638)
  
      def test_extremely_large_input_values(self):
          # Extremely large values in x and y.  These values would cause the
          # product sigma_x * sigma_y to overflow if the two factors were
          # computed independently.
          x = np.array([2.3e200, 4.5e200, 6.7e200, 8e200])
          y = np.array([1.2e199, 5.5e200, 3.3e201, 1.0e200])
          r, p = stats.pearsonr(x, y)
  
          # The expected values were computed using mpmath with 80 digits
          # of precision.
          assert_allclose(r, 0.351312332103289)
          assert_allclose(p, 0.648687667896711)
  
      def test_length_two_pos1(self):
          # Inputs with length 2.
          # See https://github.com/scipy/scipy/issues/7730
          res = stats.pearsonr([1, 2], [3, 5])
          r, p = res
          assert_equal(r, 1)
          assert_equal(p, 1)
          assert_equal(res.confidence_interval(), (-1, 1))
  
      def test_length_two_neg2(self):
          # Inputs with length 2.
          # See https://github.com/scipy/scipy/issues/7730
          r, p = stats.pearsonr([2, 1], [3, 5])
          assert_equal(r, -1)
          assert_equal(p, 1)
  
      # Expected values computed with R 3.6.2 cor.test, e.g.
      # options(digits=16)
      # x <- c(1, 2, 3, 4)
      # y <- c(0, 1, 0.5, 1)
      # cor.test(x, y, method = "pearson", alternative = "g")
      # correlation coefficient and p-value for alternative='two-sided'
      # calculated with mpmath agree to 16 digits.
      @pytest.mark.parametrize('alternative, pval, rlow, rhigh',
                               [('two-sided',
                                 0.325800137536, -0.814938968841, 0.99230697523),
                                ('less',
                                 0.8370999312316, -1, 0.985600937290653),
                                ('greater',
                                 0.1629000687684, -0.6785654158217636, 1)])
      def test_basic_example(self, alternative, pval, rlow, rhigh):
          x = [1, 2, 3, 4]
          y = [0, 1, 0.5, 1]
          result = stats.pearsonr(x, y, alternative=alternative)
          assert_allclose(result.statistic, 0.6741998624632421, rtol=1e-12)
          assert_allclose(result.pvalue, pval, rtol=1e-6)
          ci = result.confidence_interval()
          assert_allclose(ci, (rlow, rhigh), rtol=1e-6)
  
      def test_length3_r_exactly_negative_one(self):
          x = [1, 2, 3]
          y = [5, -4, -13]
          res = stats.pearsonr(x, y)
  
          # The expected r and p are exact.
          r, p = res
          assert_allclose(r, -1.0)
          assert_allclose(p, 0.0, atol=1e-7)
  
          assert_equal(res.confidence_interval(), (-1, 1))
  
      def test_unequal_lengths(self):
          x = [1, 2, 3]
          y = [4, 5]
          assert_raises(ValueError, stats.pearsonr, x, y)
  
      def test_len1(self):
          x = [1]
          y = [2]
          assert_raises(ValueError, stats.pearsonr, x, y)
  
  
  class TestFisherExact:
      """Some tests to show that fisher_exact() works correctly.
  
      Note that in SciPy 0.9.0 this was not working well for large numbers due to
      inaccuracy of the hypergeom distribution (see #1218). Fixed now.
  
      Also note that R and SciPy have different argument formats for their
      hypergeometric distribution functions.
  
      R:
      > phyper(18999, 99000, 110000, 39000, lower.tail = FALSE)
      [1] 1.701815e-09
      """
  
      def test_basic(self):
          fisher_exact = stats.fisher_exact
  
          res = fisher_exact([[14500, 20000], [30000, 40000]])[1]
          assert_approx_equal(res, 0.01106, significant=4)
          res = fisher_exact([[100, 2], [1000, 5]])[1]
          assert_approx_equal(res, 0.1301, significant=4)
          res = fisher_exact([[2, 7], [8, 2]])[1]
          assert_approx_equal(res, 0.0230141, significant=6)
          res = fisher_exact([[5, 1], [10, 10]])[1]
          assert_approx_equal(res, 0.1973244, significant=6)
          res = fisher_exact([[5, 15], [20, 20]])[1]
          assert_approx_equal(res, 0.0958044, significant=6)
          res = fisher_exact([[5, 16], [20, 25]])[1]
          assert_approx_equal(res, 0.1725862, significant=6)
          res = fisher_exact([[10, 5], [10, 1]])[1]
          assert_approx_equal(res, 0.1973244, significant=6)
          res = fisher_exact([[5, 0], [1, 4]])[1]
          assert_approx_equal(res, 0.04761904, significant=6)
          res = fisher_exact([[0, 1], [3, 2]])[1]
          assert_approx_equal(res, 1.0)
          res = fisher_exact([[0, 2], [6, 4]])[1]
          assert_approx_equal(res, 0.4545454545)
          res = fisher_exact([[2, 7], [8, 2]])
          assert_approx_equal(res[1], 0.0230141, significant=6)
          assert_approx_equal(res[0], 4.0 / 56)
  
      def test_precise(self):
          # results from R
          #
          # R defines oddsratio differently (see Notes section of fisher_exact
          # docstring), so those will not match.  We leave them in anyway, in
          # case they will be useful later on. We test only the p-value.
          tablist = [
              ([[100, 2], [1000, 5]], (2.505583993422285e-001, 1.300759363430016e-001)),
              ([[2, 7], [8, 2]], (8.586235135736206e-002, 2.301413756522114e-002)),
              ([[5, 1], [10, 10]], (4.725646047336584e+000, 1.973244147157190e-001)),
              ([[5, 15], [20, 20]], (3.394396617440852e-001, 9.580440012477637e-002)),
              ([[5, 16], [20, 25]], (3.960558326183334e-001, 1.725864953812994e-001)),
              ([[10, 5], [10, 1]], (2.116112781158483e-001, 1.973244147157190e-001)),
              ([[10, 5], [10, 0]], (0.000000000000000e+000, 6.126482213438734e-002)),
              ([[5, 0], [1, 4]], (np.inf, 4.761904761904762e-002)),
              ([[0, 5], [1, 4]], (0.000000000000000e+000, 1.000000000000000e+000)),
              ([[5, 1], [0, 4]], (np.inf, 4.761904761904758e-002)),
              ([[0, 1], [3, 2]], (0.000000000000000e+000, 1.000000000000000e+000))
              ]
          for table, res_r in tablist:
              res = stats.fisher_exact(np.asarray(table))
              np.testing.assert_almost_equal(res[1], res_r[1], decimal=11,
                                             verbose=True)
  
      def test_gh4130(self):
          # Previously, a fudge factor used to distinguish between theoeretically
          # and numerically different probability masses was 1e-4; it has been
          # tightened to fix gh4130. Accuracy checked against R fisher.test.
          # options(digits=16)
          # table <- matrix(c(6, 108, 37, 200), nrow = 2)
          # fisher.test(table, alternative = "t")
          x = [[6, 37], [108, 200]]
          res = stats.fisher_exact(x)
          assert_allclose(res[1], 0.005092697748126)
  
          # case from https://github.com/brentp/fishers_exact_test/issues/27
          # That package has an (absolute?) fudge factor of 1e-6; too big
          x = [[22, 0], [0, 102]]
          res = stats.fisher_exact(x)
          assert_allclose(res[1], 7.175066786244549e-25)
  
          # case from https://github.com/brentp/fishers_exact_test/issues/1
          x = [[94, 48], [3577, 16988]]
          res = stats.fisher_exact(x)
          assert_allclose(res[1], 2.069356340993818e-37)
  
      def test_gh9231(self):
          # Previously, fisher_exact was extremely slow for this table
          # As reported in gh-9231, the p-value should be very nearly zero
          x = [[5829225, 5692693], [5760959, 5760959]]
          res = stats.fisher_exact(x)
          assert_allclose(res[1], 0, atol=1e-170)
  
      @pytest.mark.slow
      def test_large_numbers(self):
          # Test with some large numbers. Regression test for #1401
          pvals = [5.56e-11, 2.666e-11, 1.363e-11]  # from R
          for pval, num in zip(pvals, [75, 76, 77]):
              res = stats.fisher_exact([[17704, 496], [1065, num]])[1]
              assert_approx_equal(res, pval, significant=4)
  
          res = stats.fisher_exact([[18000, 80000], [20000, 90000]])[1]
          assert_approx_equal(res, 0.2751, significant=4)
  
      def test_raises(self):
          # test we raise an error for wrong shape of input.
          assert_raises(ValueError, stats.fisher_exact,
                        np.arange(6).reshape(2, 3))
  
      def test_row_or_col_zero(self):
          tables = ([[0, 0], [5, 10]],
                    [[5, 10], [0, 0]],
                    [[0, 5], [0, 10]],
                    [[5, 0], [10, 0]])
          for table in tables:
              oddsratio, pval = stats.fisher_exact(table)
              assert_equal(pval, 1.0)
              assert_equal(oddsratio, np.nan)
  
      def test_less_greater(self):
          tables = (
              # Some tables to compare with R:
              [[2, 7], [8, 2]],
              [[200, 7], [8, 300]],
              [[28, 21], [6, 1957]],
              [[190, 800], [200, 900]],
              # Some tables with simple exact values
              # (includes regression test for ticket #1568):
              [[0, 2], [3, 0]],
              [[1, 1], [2, 1]],
              [[2, 0], [1, 2]],
              [[0, 1], [2, 3]],
              [[1, 0], [1, 4]],
              )
          pvals = (
              # from R:
              [0.018521725952066501, 0.9990149169715733],
              [1.0, 2.0056578803889148e-122],
              [1.0, 5.7284374608319831e-44],
              [0.7416227, 0.2959826],
              # Exact:
              [0.1, 1.0],
              [0.7, 0.9],
              [1.0, 0.3],
              [2./3, 1.0],
              [1.0, 1./3],
              )
          for table, pval in zip(tables, pvals):
              res = []
              res.append(stats.fisher_exact(table, alternative="less")[1])
              res.append(stats.fisher_exact(table, alternative="greater")[1])
              assert_allclose(res, pval, atol=0, rtol=1e-7)
  
      def test_gh3014(self):
          # check if issue #3014 has been fixed.
          # before, this would have risen a ValueError
          odds, pvalue = stats.fisher_exact([[1, 2], [9, 84419233]])
  
  
  class TestCorrSpearmanr:
      """ W.II.D. Compute a correlation matrix on all the variables.
  
          All the correlations, except for ZERO and MISS, should be exactly 1.
          ZERO and MISS should have undefined or missing correlations with the
          other variables.  The same should go for SPEARMAN corelations, if
          your program has them.
      """
  
      def test_scalar(self):
          y = stats.spearmanr(4., 2.)
          assert_(np.isnan(y).all())
  
      def test_uneven_lengths(self):
          assert_raises(ValueError, stats.spearmanr, [1, 2, 1], [8, 9])
          assert_raises(ValueError, stats.spearmanr, [1, 2, 1], 8)
  
      def test_uneven_2d_shapes(self):
          # Different number of columns should work - those just get concatenated.
          np.random.seed(232324)
          x = np.random.randn(4, 3)
          y = np.random.randn(4, 2)
          assert stats.spearmanr(x, y).correlation.shape == (5, 5)
          assert stats.spearmanr(x.T, y.T, axis=1).pvalue.shape == (5, 5)
  
          assert_raises(ValueError, stats.spearmanr, x, y, axis=1)
          assert_raises(ValueError, stats.spearmanr, x.T, y.T)
  
      def test_ndim_too_high(self):
          np.random.seed(232324)
          x = np.random.randn(4, 3, 2)
          assert_raises(ValueError, stats.spearmanr, x)
          assert_raises(ValueError, stats.spearmanr, x, x)
          assert_raises(ValueError, stats.spearmanr, x, None, None)
          # But should work with axis=None (raveling axes) for two input arrays
          assert_allclose(stats.spearmanr(x, x, axis=None),
                          stats.spearmanr(x.flatten(), x.flatten(), axis=0))
  
      def test_nan_policy(self):
          x = np.arange(10.)
          x[9] = np.nan
          assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
          assert_array_equal(stats.spearmanr(x, x, nan_policy='omit'),
                             (1.0, 0.0))
          assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
          assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')
  
      def test_nan_policy_bug_12458(self):
          np.random.seed(5)
          x = np.random.rand(5, 10)
          k = 6
          x[:, k] = np.nan
          y = np.delete(x, k, axis=1)
          corx, px = stats.spearmanr(x, nan_policy='omit')
          cory, py = stats.spearmanr(y)
          corx = np.delete(np.delete(corx, k, axis=1), k, axis=0)
          px = np.delete(np.delete(px, k, axis=1), k, axis=0)
          assert_allclose(corx, cory, atol=1e-14)
          assert_allclose(px, py, atol=1e-14)
  
      def test_nan_policy_bug_12411(self):
          np.random.seed(5)
          m = 5
          n = 10
          x = np.random.randn(m, n)
          x[1, 0] = np.nan
          x[3, -1] = np.nan
          corr, pvalue = stats.spearmanr(x, axis=1, nan_policy="propagate")
          res = [[stats.spearmanr(x[i, :], x[j, :]).correlation for i in range(m)]
                 for j in range(m)]
          assert_allclose(corr, res)
  
      def test_sXX(self):
          y = stats.spearmanr(X,X)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sXBIG(self):
          y = stats.spearmanr(X,BIG)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sXLITTLE(self):
          y = stats.spearmanr(X,LITTLE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sXHUGE(self):
          y = stats.spearmanr(X,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sXTINY(self):
          y = stats.spearmanr(X,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sXROUND(self):
          y = stats.spearmanr(X,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sBIGBIG(self):
          y = stats.spearmanr(BIG,BIG)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sBIGLITTLE(self):
          y = stats.spearmanr(BIG,LITTLE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sBIGHUGE(self):
          y = stats.spearmanr(BIG,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sBIGTINY(self):
          y = stats.spearmanr(BIG,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sBIGROUND(self):
          y = stats.spearmanr(BIG,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sLITTLELITTLE(self):
          y = stats.spearmanr(LITTLE,LITTLE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sLITTLEHUGE(self):
          y = stats.spearmanr(LITTLE,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sLITTLETINY(self):
          y = stats.spearmanr(LITTLE,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sLITTLEROUND(self):
          y = stats.spearmanr(LITTLE,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sHUGEHUGE(self):
          y = stats.spearmanr(HUGE,HUGE)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sHUGETINY(self):
          y = stats.spearmanr(HUGE,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sHUGEROUND(self):
          y = stats.spearmanr(HUGE,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sTINYTINY(self):
          y = stats.spearmanr(TINY,TINY)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sTINYROUND(self):
          y = stats.spearmanr(TINY,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_sROUNDROUND(self):
          y = stats.spearmanr(ROUND,ROUND)
          r = y[0]
          assert_approx_equal(r,1.0)
  
      def test_spearmanr_result_attributes(self):
          res = stats.spearmanr(X, X)
          attributes = ('correlation', 'pvalue')
          check_named_results(res, attributes)
  
      def test_1d_vs_2d(self):
          x1 = [1, 2, 3, 4, 5, 6]
          x2 = [1, 2, 3, 4, 6, 5]
          res1 = stats.spearmanr(x1, x2)
          res2 = stats.spearmanr(np.asarray([x1, x2]).T)
          assert_allclose(res1, res2)
  
      def test_1d_vs_2d_nans(self):
          # Now the same with NaNs present.  Regression test for gh-9103.
          for nan_policy in ['propagate', 'omit']:
              x1 = [1, np.nan, 3, 4, 5, 6]
              x2 = [1, 2, 3, 4, 6, np.nan]
              res1 = stats.spearmanr(x1, x2, nan_policy=nan_policy)
              res2 = stats.spearmanr(np.asarray([x1, x2]).T, nan_policy=nan_policy)
              assert_allclose(res1, res2)
  
      def test_3cols(self):
          x1 = np.arange(6)
          x2 = -x1
          x3 = np.array([0, 1, 2, 3, 5, 4])
          x = np.asarray([x1, x2, x3]).T
          actual = stats.spearmanr(x)
          expected_corr = np.array([[1, -1, 0.94285714],
                                    [-1, 1, -0.94285714],
                                    [0.94285714, -0.94285714, 1]])
          expected_pvalue = np.zeros((3, 3), dtype=float)
          expected_pvalue[2, 0:2] = 0.00480466472
          expected_pvalue[0:2, 2] = 0.00480466472
  
          assert_allclose(actual.correlation, expected_corr)
          assert_allclose(actual.pvalue, expected_pvalue)
  
      def test_gh_9103(self):
          # Regression test for gh-9103.
          x = np.array([[np.nan, 3.0, 4.0, 5.0, 5.1, 6.0, 9.2],
                        [5.0, np.nan, 4.1, 4.8, 4.9, 5.0, 4.1],
                        [0.5, 4.0, 7.1, 3.8, 8.0, 5.1, 7.6]]).T
          corr = np.array([[np.nan, np.nan, np.nan],
                           [np.nan, np.nan, np.nan],
                           [np.nan, np.nan, 1.]])
          assert_allclose(stats.spearmanr(x, nan_policy='propagate').correlation,
                          corr)
  
          res = stats.spearmanr(x, nan_policy='omit').correlation
          assert_allclose((res[0][1], res[0][2], res[1][2]),
                          (0.2051957, 0.4857143, -0.4707919), rtol=1e-6)
  
      def test_gh_8111(self):
          # Regression test for gh-8111 (different result for float/int/bool).
          n = 100
          np.random.seed(234568)
          x = np.random.rand(n)
          m = np.random.rand(n) > 0.7
  
          # bool against float, no nans
          a = (x > .5)
          b = np.array(x)
          res1 = stats.spearmanr(a, b, nan_policy='omit').correlation
  
          # bool against float with NaNs
          b[m] = np.nan
          res2 = stats.spearmanr(a, b, nan_policy='omit').correlation
  
          # int against float with NaNs
          a = a.astype(np.int32)
          res3 = stats.spearmanr(a, b, nan_policy='omit').correlation
  
          expected = [0.865895477, 0.866100381, 0.866100381]
          assert_allclose([res1, res2, res3], expected)
  
  
  class TestCorrSpearmanr2:
      """Some further tests of the spearmanr function."""
  
      def test_spearmanr_vs_r(self):
          # Cross-check with R:
          # cor.test(c(1,2,3,4,5),c(5,6,7,8,7),method="spearmanr")
          x1 = [1, 2, 3, 4, 5]
          x2 = [5, 6, 7, 8, 7]
          expected = (0.82078268166812329, 0.088587005313543798)
          res = stats.spearmanr(x1, x2)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      def test_empty_arrays(self):
          assert_equal(stats.spearmanr([], []), (np.nan, np.nan))
  
      def test_normal_draws(self):
          np.random.seed(7546)
          x = np.array([np.random.normal(loc=1, scale=1, size=500),
                      np.random.normal(loc=1, scale=1, size=500)])
          corr = [[1.0, 0.3],
                  [0.3, 1.0]]
          x = np.dot(np.linalg.cholesky(corr), x)
          expected = (0.28659685838743354, 6.579862219051161e-11)
          res = stats.spearmanr(x[0], x[1])
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      def test_corr_1(self):
          assert_approx_equal(stats.spearmanr([1, 1, 2], [1, 1, 2])[0], 1.0)
  
      def test_nan_policies(self):
          x = np.arange(10.)
          x[9] = np.nan
          assert_array_equal(stats.spearmanr(x, x), (np.nan, np.nan))
          assert_allclose(stats.spearmanr(x, x, nan_policy='omit'),
                          (1.0, 0))
          assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='raise')
          assert_raises(ValueError, stats.spearmanr, x, x, nan_policy='foobar')
  
      def test_unequal_lengths(self):
          x = np.arange(10.)
          y = np.arange(20.)
          assert_raises(ValueError, stats.spearmanr, x, y)
  
      def test_omit_paired_value(self):
          x1 = [1, 2, 3, 4]
          x2 = [8, 7, 6, np.nan]
          res1 = stats.spearmanr(x1, x2, nan_policy='omit')
          res2 = stats.spearmanr(x1[:3], x2[:3], nan_policy='omit')
          assert_equal(res1, res2)
  
      def test_gh_issue_6061_windows_overflow(self):
          x = list(range(2000))
          y = list(range(2000))
          y[0], y[9] = y[9], y[0]
          y[10], y[434] = y[434], y[10]
          y[435], y[1509] = y[1509], y[435]
          # rho = 1 - 6 * (2 * (9^2 + 424^2 + 1074^2))/(2000 * (2000^2 - 1))
          #     = 1 - (1 / 500)
          #     = 0.998
          x.append(np.nan)
          y.append(3.0)
          assert_almost_equal(stats.spearmanr(x, y, nan_policy='omit')[0], 0.998)
  
      def test_tie0(self):
          # with only ties in one or both inputs
          with assert_warns(stats.SpearmanRConstantInputWarning):
              r, p = stats.spearmanr([2, 2, 2], [2, 2, 2])
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
              r, p = stats.spearmanr([2, 0, 2], [2, 2, 2])
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
              r, p = stats.spearmanr([2, 2, 2], [2, 0, 2])
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
  
      def test_tie1(self):
          # Data
          x = [1.0, 2.0, 3.0, 4.0]
          y = [1.0, 2.0, 2.0, 3.0]
          # Ranks of the data, with tie-handling.
          xr = [1.0, 2.0, 3.0, 4.0]
          yr = [1.0, 2.5, 2.5, 4.0]
          # Result of spearmanr should be the same as applying
          # pearsonr to the ranks.
          sr = stats.spearmanr(x, y)
          pr = stats.pearsonr(xr, yr)
          assert_almost_equal(sr, pr)
  
      def test_tie2(self):
          # Test tie-handling if inputs contain nan's
          # Data without nan's
          x1 = [1, 2, 2.5, 2]
          y1 = [1, 3, 2.5, 4]
          # Same data with nan's
          x2 = [1, 2, 2.5, 2, np.nan]
          y2 = [1, 3, 2.5, 4, np.nan]
  
          # Results for two data sets should be the same if nan's are ignored
          sr1 = stats.spearmanr(x1, y1)
          sr2 = stats.spearmanr(x2, y2, nan_policy='omit')
          assert_almost_equal(sr1, sr2)
  
      def test_ties_axis_1(self):
          z1 = np.array([[1, 1, 1, 1], [1, 2, 3, 4]])
          z2 = np.array([[1, 2, 3, 4], [1, 1, 1, 1]])
          z3 = np.array([[1, 1, 1, 1], [1, 1, 1, 1]])
          with assert_warns(stats.SpearmanRConstantInputWarning):
              r, p = stats.spearmanr(z1, axis=1)
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
              r, p = stats.spearmanr(z2, axis=1)
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
              r, p = stats.spearmanr(z3, axis=1)
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
  
      def test_gh_11111(self):
          x = np.array([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
          y = np.array([0, 0.009783728115345005, 0, 0, 0.0019759230121848587,
              0.0007535430349118562, 0.0002661781514710257, 0, 0,
              0.0007835762419683435])
          with assert_warns(stats.SpearmanRConstantInputWarning):
              r, p = stats.spearmanr(x, y)
              assert_equal(r, np.nan)
              assert_equal(p, np.nan)
  
      def test_index_error(self):
          x = np.array([1.0, 7.0, 2.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
          y = np.array([0, 0.009783728115345005, 0, 0, 0.0019759230121848587,
              0.0007535430349118562, 0.0002661781514710257, 0, 0,
              0.0007835762419683435])
          assert_raises(ValueError, stats.spearmanr, x, y, axis=2)
  
      def test_alternative(self):
          # Test alternative parameter
  
          # Simple test - Based on the above ``test_spearmanr_vs_r``
          x1 = [1, 2, 3, 4, 5]
          x2 = [5, 6, 7, 8, 7]
  
          # strong positive correlation
          expected = (0.82078268166812329, 0.088587005313543798)
  
          # correlation > 0 -> large "less" p-value
          res = stats.spearmanr(x1, x2, alternative="less")
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], 1 - (expected[1] / 2))
  
          # correlation > 0 -> small "less" p-value
          res = stats.spearmanr(x1, x2, alternative="greater")
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1] / 2)
  
          with pytest.raises(ValueError, match="alternative must be 'less'..."):
              stats.spearmanr(x1, x2, alternative="ekki-ekki")
  
      @pytest.mark.parametrize("alternative", ('two-sided', 'less', 'greater'))
      def test_alternative_nan_policy(self, alternative):
          # Test nan policies
          x1 = [1, 2, 3, 4, 5]
          x2 = [5, 6, 7, 8, 7]
          x1nan = x1 + [np.nan]
          x2nan = x2 + [np.nan]
  
          # test nan_policy="propagate"
          assert_array_equal(stats.spearmanr(x1nan, x2nan), (np.nan, np.nan))
  
          # test nan_policy="omit"
          res_actual = stats.spearmanr(x1nan, x2nan, nan_policy='omit',
                                       alternative=alternative)
          res_expected = stats.spearmanr(x1, x2, alternative=alternative)
          assert_allclose(res_actual, res_expected)
  
          # test nan_policy="raise"
          message = 'The input contains nan values'
          with pytest.raises(ValueError, match=message):
              stats.spearmanr(x1nan, x2nan, nan_policy='raise',
                              alternative=alternative)
  
          # test invalid nan_policy
          message = "nan_policy must be one of..."
          with pytest.raises(ValueError, match=message):
              stats.spearmanr(x1nan, x2nan, nan_policy='ekki-ekki',
                              alternative=alternative)
  
  
  #    W.II.E.  Tabulate X against X, using BIG as a case weight.  The values
  #    should appear on the diagonal and the total should be 899999955.
  #    If the table cannot hold these values, forget about working with
  #    census data.  You can also tabulate HUGE against TINY.  There is no
  #    reason a tabulation program should not be able to distinguish
  #    different values regardless of their magnitude.
  
  # I need to figure out how to do this one.
  
  
  def test_kendalltau():
      # For the cases without ties, both variants should give the same
      # result.
      variants = ('b', 'c')
  
      # case without ties, con-dis equal zero
      x = [5, 2, 1, 3, 6, 4, 7, 8]
      y = [5, 2, 6, 3, 1, 8, 7, 4]
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (0.0, 1.0)
      for taux in variants:
          res = stats.kendalltau(x, y)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # case without ties, con-dis equal zero
      x = [0, 5, 2, 1, 3, 6, 4, 7, 8]
      y = [5, 2, 0, 6, 3, 1, 8, 7, 4]
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (0.0, 1.0)
      for taux in variants:
          res = stats.kendalltau(x, y)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # case without ties, con-dis close to zero
      x = [5, 2, 1, 3, 6, 4, 7]
      y = [5, 2, 6, 3, 1, 7, 4]
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (-0.14285714286, 0.77261904762)
      for taux in variants:
          res = stats.kendalltau(x, y)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # case without ties, con-dis close to zero
      x = [2, 1, 3, 6, 4, 7, 8]
      y = [2, 6, 3, 1, 8, 7, 4]
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (0.047619047619, 1.0)
      for taux in variants:
          res = stats.kendalltau(x, y)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # simple case without ties
      x = np.arange(10)
      y = np.arange(10)
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (1.0, 5.511463844797e-07)
      for taux in variants:
          res = stats.kendalltau(x, y, variant=taux)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # swap a couple of values
      b = y[1]
      y[1] = y[2]
      y[2] = b
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (0.9555555555555556, 5.511463844797e-06)
      for taux in variants:
          res = stats.kendalltau(x, y, variant=taux)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # swap a couple more
      b = y[5]
      y[5] = y[6]
      y[6] = b
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (0.9111111111111111, 2.976190476190e-05)
      for taux in variants:
          res = stats.kendalltau(x, y, variant=taux)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # same in opposite direction
      x = np.arange(10)
      y = np.arange(10)[::-1]
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (-1.0, 5.511463844797e-07)
      for taux in variants:
          res = stats.kendalltau(x, y, variant=taux)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # swap a couple of values
      b = y[1]
      y[1] = y[2]
      y[2] = b
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (-0.9555555555555556, 5.511463844797e-06)
      for taux in variants:
          res = stats.kendalltau(x, y, variant=taux)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # swap a couple more
      b = y[5]
      y[5] = y[6]
      y[6] = b
      # Cross-check with exact result from R:
      # cor.test(x,y,method="kendall",exact=1)
      expected = (-0.9111111111111111, 2.976190476190e-05)
      for taux in variants:
          res = stats.kendalltau(x, y, variant=taux)
          assert_approx_equal(res[0], expected[0])
          assert_approx_equal(res[1], expected[1])
  
      # Check a case where variants are different
      # Example values found from Kendall (1970).
      # P-value is the same for the both variants
      x = array([1, 2, 2, 4, 4, 6, 6, 8, 9, 9])
      y = array([1, 2, 4, 4, 4, 4, 8, 8, 8, 10])
      expected = 0.85895569
      assert_approx_equal(stats.kendalltau(x, y, variant='b')[0], expected)
      expected = 0.825
      assert_approx_equal(stats.kendalltau(x, y, variant='c')[0], expected)
  
      # check exception in case of ties and method='exact' requested
      y[2] = y[1]
      assert_raises(ValueError, stats.kendalltau, x, y, method='exact')
  
      # check exception in case of invalid method keyword
      assert_raises(ValueError, stats.kendalltau, x, y, method='banana')
  
      # check exception in case of invalid variant keyword
      assert_raises(ValueError, stats.kendalltau, x, y, variant='rms')
  
      # tau-b with some ties
      # Cross-check with R:
      # cor.test(c(12,2,1,12,2),c(1,4,7,1,0),method="kendall",exact=FALSE)
      x1 = [12, 2, 1, 12, 2]
      x2 = [1, 4, 7, 1, 0]
      expected = (-0.47140452079103173, 0.28274545993277478)
      res = stats.kendalltau(x1, x2)
      assert_approx_equal(res[0], expected[0])
      assert_approx_equal(res[1], expected[1])
  
      # test for namedtuple attribute results
      attributes = ('correlation', 'pvalue')
      for taux in variants:
          res = stats.kendalltau(x1, x2, variant=taux)
          check_named_results(res, attributes)
  
      # with only ties in one or both inputs in tau-b or tau-c
      for taux in variants:
          assert_equal(stats.kendalltau([2, 2, 2], [2, 2, 2], variant=taux),
                       (np.nan, np.nan))
          assert_equal(stats.kendalltau([2, 0, 2], [2, 2, 2], variant=taux),
                       (np.nan, np.nan))
          assert_equal(stats.kendalltau([2, 2, 2], [2, 0, 2], variant=taux),
                       (np.nan, np.nan))
  
      # empty arrays provided as input
      assert_equal(stats.kendalltau([], []), (np.nan, np.nan))
  
      # check with larger arrays
      np.random.seed(7546)
      x = np.array([np.random.normal(loc=1, scale=1, size=500),
                  np.random.normal(loc=1, scale=1, size=500)])
      corr = [[1.0, 0.3],
              [0.3, 1.0]]
      x = np.dot(np.linalg.cholesky(corr), x)
      expected = (0.19291382765531062, 1.1337095377742629e-10)
      res = stats.kendalltau(x[0], x[1])
      assert_approx_equal(res[0], expected[0])
      assert_approx_equal(res[1], expected[1])
  
      # this should result in 1 for taub but not tau-c
      assert_approx_equal(stats.kendalltau([1, 1, 2], [1, 1, 2], variant='b')[0],
                          1.0)
      assert_approx_equal(stats.kendalltau([1, 1, 2], [1, 1, 2], variant='c')[0],
                          0.88888888)
  
      # test nan_policy
      x = np.arange(10.)
      x[9] = np.nan
      assert_array_equal(stats.kendalltau(x, x), (np.nan, np.nan))
      assert_allclose(stats.kendalltau(x, x, nan_policy='omit'),
                      (1.0, 5.5114638e-6), rtol=1e-06)
      assert_allclose(stats.kendalltau(x, x, nan_policy='omit', method='asymptotic'),
                      (1.0, 0.00017455009626808976), rtol=1e-06)
      assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='raise')
      assert_raises(ValueError, stats.kendalltau, x, x, nan_policy='foobar')
  
      # test unequal length inputs
      x = np.arange(10.)
      y = np.arange(20.)
      assert_raises(ValueError, stats.kendalltau, x, y)
  
      # test all ties
      tau, p_value = stats.kendalltau([], [])
      assert_equal(np.nan, tau)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.kendalltau([0], [0])
      assert_equal(np.nan, tau)
      assert_equal(np.nan, p_value)
  
      # Regression test for GitHub issue #6061 - Overflow on Windows
      x = np.arange(2000, dtype=float)
      x = np.ma.masked_greater(x, 1995)
      y = np.arange(2000, dtype=float)
      y = np.concatenate((y[1000:], y[:1000]))
      assert_(np.isfinite(stats.kendalltau(x,y)[1]))
  
  
  def test_kendalltau_vs_mstats_basic():
      np.random.seed(42)
      for s in range(2,10):
          a = []
          # Generate rankings with ties
          for i in range(s):
              a += [i]*i
          b = list(a)
          np.random.shuffle(a)
          np.random.shuffle(b)
          expected = mstats_basic.kendalltau(a, b)
          actual = stats.kendalltau(a, b)
          assert_approx_equal(actual[0], expected[0])
          assert_approx_equal(actual[1], expected[1])
  
  
  def test_kendalltau_nan_2nd_arg():
      # regression test for gh-6134: nans in the second arg were not handled
      x = [1., 2., 3., 4.]
      y = [np.nan, 2.4, 3.4, 3.4]
  
      r1 = stats.kendalltau(x, y, nan_policy='omit')
      r2 = stats.kendalltau(x[1:], y[1:])
      assert_allclose(r1.correlation, r2.correlation, atol=1e-15)
  
  
  class TestKendallTauAlternative:
      def test_kendalltau_alternative_asymptotic(self):
          # Test alternative parameter, asymptotic method (due to tie)
  
          # Based on TestCorrSpearman2::test_alternative
          x1 = [1, 2, 3, 4, 5]
          x2 = [5, 6, 7, 8, 7]
  
          # strong positive correlation
          expected = stats.kendalltau(x1, x2, alternative="two-sided")
          assert expected[0] > 0
  
          # rank correlation > 0 -> large "less" p-value
          res = stats.kendalltau(x1, x2, alternative="less")
          assert_equal(res[0], expected[0])
          assert_allclose(res[1], 1 - (expected[1] / 2))
  
          # rank correlation > 0 -> small "greater" p-value
          res = stats.kendalltau(x1, x2, alternative="greater")
          assert_equal(res[0], expected[0])
          assert_allclose(res[1], expected[1] / 2)
  
          # reverse the direction of rank correlation
          x2.reverse()
  
          # strong negative correlation
          expected = stats.kendalltau(x1, x2, alternative="two-sided")
          assert expected[0] < 0
  
          # rank correlation < 0 -> large "greater" p-value
          res = stats.kendalltau(x1, x2, alternative="greater")
          assert_equal(res[0], expected[0])
          assert_allclose(res[1], 1 - (expected[1] / 2))
  
          # rank correlation < 0 -> small "less" p-value
          res = stats.kendalltau(x1, x2, alternative="less")
          assert_equal(res[0], expected[0])
          assert_allclose(res[1], expected[1] / 2)
  
          with pytest.raises(ValueError, match="alternative must be 'less'..."):
              stats.kendalltau(x1, x2, alternative="ekki-ekki")
  
      # There are a lot of special cases considered in the calculation of the
      # exact p-value, so we test each separately. We also need to test
      # separately when the observed statistic is in the left tail vs the right
      # tail because the code leverages symmetry of the null distribution; to
      # do that we use the same test case but negate one of the samples.
      # Reference values computed using R cor.test, e.g.
      # options(digits=16)
      # x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
      # y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
      # cor.test(x, y, method = "kendall", alternative = "g")
  
      alternatives = ('less', 'two-sided', 'greater')
      p_n1 = [np.nan, np.nan, np.nan]
      p_n2 = [1, 1, 0.5]
      p_c0 = [1, 0.3333333333333, 0.1666666666667]
      p_c1 = [0.9583333333333, 0.3333333333333, 0.1666666666667]
      p_no_correlation = [0.5916666666667, 1, 0.5916666666667]
      p_no_correlationb = [0.5475694444444, 1, 0.5475694444444]
      p_n_lt_171 = [0.9624118165785, 0.1194389329806, 0.0597194664903]
      p_n_lt_171b = [0.246236925303, 0.4924738506059, 0.755634083327]
      p_n_lt_171c = [0.9847475308925, 0.03071385306533, 0.01535692653267]
  
      def exact_test(self, x, y, alternative, rev, stat_expected, p_expected):
          if rev:
              y = -np.asarray(y)
              stat_expected *= -1
          res = stats.kendalltau(x, y, method='exact', alternative=alternative)
          res_expected = stat_expected, p_expected
          assert_allclose(res, res_expected)
  
      case_R_n1 = (list(zip(alternatives, p_n1, [False]*3))
                   + list(zip(alternatives, reversed(p_n1), [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_n1)
      def test_against_R_n1(self, alternative, p_expected, rev):
          x, y = [1], [2]
          stat_expected = np.nan
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_n2 = (list(zip(alternatives, p_n2, [False]*3))
                   + list(zip(alternatives, reversed(p_n2), [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_n2)
      def test_against_R_n2(self, alternative, p_expected, rev):
          x, y = [1, 2], [3, 4]
          stat_expected = 0.9999999999999998
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_c0 = (list(zip(alternatives, p_c0, [False]*3))
                   + list(zip(alternatives, reversed(p_c0), [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_c0)
      def test_against_R_c0(self, alternative, p_expected, rev):
          x, y = [1, 2, 3], [1, 2, 3]
          stat_expected = 1
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_c1 = (list(zip(alternatives, p_c1, [False]*3))
                   + list(zip(alternatives, reversed(p_c1), [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_c1)
      def test_against_R_c1(self, alternative, p_expected, rev):
          x, y = [1, 2, 3, 4], [1, 2, 4, 3]
          stat_expected = 0.6666666666666667
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_no_corr = (list(zip(alternatives, p_no_correlation, [False]*3))
                        + list(zip(alternatives, reversed(p_no_correlation),
                                   [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_no_corr)
      def test_against_R_no_correlation(self, alternative, p_expected, rev):
          x, y = [1, 2, 3, 4, 5], [1, 5, 4, 2, 3]
          stat_expected = 0
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_no_cor_b = (list(zip(alternatives, p_no_correlationb, [False]*3))
                       + list(zip(alternatives, reversed(p_no_correlationb),
                                  [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_no_cor_b)
      def test_against_R_no_correlationb(self, alternative, p_expected, rev):
          x, y = [1, 2, 3, 4, 5, 6, 7, 8], [8, 6, 1, 3, 2, 5, 4, 7]
          stat_expected = 0
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_lt_171 = (list(zip(alternatives, p_n_lt_171, [False]*3))
                       + list(zip(alternatives, reversed(p_n_lt_171), [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_lt_171)
      def test_against_R_lt_171(self, alternative, p_expected, rev):
          # Data from Hollander & Wolfe (1973), p. 187f.
          # Used from https://rdrr.io/r/stats/cor.test.html
          x = [44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1]
          y = [2.6, 3.1, 2.5, 5.0, 3.6, 4.0, 5.2, 2.8, 3.8]
          stat_expected = 0.4444444444444445
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_lt_171b = (list(zip(alternatives, p_n_lt_171b, [False]*3))
                        + list(zip(alternatives, reversed(p_n_lt_171b),
                                   [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_lt_171b)
      def test_against_R_lt_171b(self, alternative, p_expected, rev):
          np.random.seed(0)
          x = np.random.rand(100)
          y = np.random.rand(100)
          stat_expected = -0.04686868686868687
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_R_lt_171c = (list(zip(alternatives, p_n_lt_171c, [False]*3))
                        + list(zip(alternatives, reversed(p_n_lt_171c),
                                   [True]*3)))
  
      @pytest.mark.parametrize("alternative, p_expected, rev", case_R_lt_171c)
      def test_against_R_lt_171c(self, alternative, p_expected, rev):
          np.random.seed(0)
          x = np.random.rand(170)
          y = np.random.rand(170)
          stat_expected = 0.1115906717716673
          self.exact_test(x, y, alternative, rev, stat_expected, p_expected)
  
      case_gt_171 = (list(zip(alternatives, [False]*3)) +
                     list(zip(alternatives, [True]*3)))
  
      @pytest.mark.parametrize("alternative, rev", case_gt_171)
      def test_gt_171(self, alternative, rev):
          np.random.seed(0)
          x = np.random.rand(400)
          y = np.random.rand(400)
          res0 = stats.kendalltau(x, y, method='exact',
                                  alternative=alternative)
          res1 = stats.kendalltau(x, y, method='asymptotic',
                                  alternative=alternative)
          assert_equal(res0[0], res1[0])
          assert_allclose(res0[1], res1[1], rtol=1e-3)
  
      @pytest.mark.parametrize("method", ('exact', 'asymptotic'))
      @pytest.mark.parametrize("alternative", ('two-sided', 'less', 'greater'))
      def test_nan_policy(self, method, alternative):
          # Test nan policies
          x1 = [1, 2, 3, 4, 5]
          x2 = [5, 6, 7, 8, 9]
          x1nan = x1 + [np.nan]
          x2nan = x2 + [np.nan]
  
          # test nan_policy="propagate"
          res_actual = stats.kendalltau(x1nan, x2nan,
                                        method=method, alternative=alternative)
          res_expected = (np.nan, np.nan)
          assert_allclose(res_actual, res_expected)
  
          # test nan_policy="omit"
          res_actual = stats.kendalltau(x1nan, x2nan, nan_policy='omit',
                                        method=method, alternative=alternative)
          res_expected = stats.kendalltau(x1, x2, method=method,
                                          alternative=alternative)
          assert_allclose(res_actual, res_expected)
  
          # test nan_policy="raise"
          message = 'The input contains nan values'
          with pytest.raises(ValueError, match=message):
              stats.kendalltau(x1nan, x2nan, nan_policy='raise',
                               method=method, alternative=alternative)
  
          # test invalid nan_policy
          message = "nan_policy must be one of..."
          with pytest.raises(ValueError, match=message):
              stats.kendalltau(x1nan, x2nan, nan_policy='ekki-ekki',
                               method=method, alternative=alternative)
  
  
  def test_weightedtau():
      x = [12, 2, 1, 12, 2]
      y = [1, 4, 7, 1, 0]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.56694968153682723)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau(x, y, additive=False)
      assert_approx_equal(tau, -0.62205716951801038)
      assert_equal(np.nan, p_value)
      # This must be exactly Kendall's tau
      tau, p_value = stats.weightedtau(x, y, weigher=lambda x: 1)
      assert_approx_equal(tau, -0.47140452079103173)
      assert_equal(np.nan, p_value)
  
      # Asymmetric, ranked version
      tau, p_value = stats.weightedtau(x, y, rank=None)
      assert_approx_equal(tau, -0.4157652301037516)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau(y, x, rank=None)
      assert_approx_equal(tau, -0.7181341329699029)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau(x, y, rank=None, additive=False)
      assert_approx_equal(tau, -0.40644850966246893)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau(y, x, rank=None, additive=False)
      assert_approx_equal(tau, -0.83766582937355172)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau(x, y, rank=False)
      assert_approx_equal(tau, -0.51604397940261848)
      assert_equal(np.nan, p_value)
      # This must be exactly Kendall's tau
      tau, p_value = stats.weightedtau(x, y, rank=True, weigher=lambda x: 1)
      assert_approx_equal(tau, -0.47140452079103173)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau(y, x, rank=True, weigher=lambda x: 1)
      assert_approx_equal(tau, -0.47140452079103173)
      assert_equal(np.nan, p_value)
      # Test argument conversion
      tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), y)
      assert_approx_equal(tau, -0.56694968153682723)
      tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.int16), y)
      assert_approx_equal(tau, -0.56694968153682723)
      tau, p_value = stats.weightedtau(np.asarray(x, dtype=np.float64), np.asarray(y, dtype=np.float64))
      assert_approx_equal(tau, -0.56694968153682723)
      # All ties
      tau, p_value = stats.weightedtau([], [])
      assert_equal(np.nan, tau)
      assert_equal(np.nan, p_value)
      tau, p_value = stats.weightedtau([0], [0])
      assert_equal(np.nan, tau)
      assert_equal(np.nan, p_value)
      # Size mismatches
      assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1, 2])
      assert_raises(ValueError, stats.weightedtau, [0, 1], [0, 1], [0])
      # NaNs
      x = [12, 2, 1, 12, 2]
      y = [1, 4, 7, 1, np.nan]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.56694968153682723)
      x = [12, 2, np.nan, 12, 2]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.56694968153682723)
      # NaNs when the dtype of x and y are all np.float64
      x = [12.0, 2.0, 1.0, 12.0, 2.0]
      y = [1.0, 4.0, 7.0, 1.0, np.nan]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.56694968153682723)
      x = [12.0, 2.0, np.nan, 12.0, 2.0]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.56694968153682723)
      # NaNs when there are more than one NaN in x or y
      x = [12.0, 2.0, 1.0, 12.0, 1.0]
      y = [1.0, 4.0, 7.0, 1.0, 1.0]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.6615242347139803)
      x = [12.0, 2.0, np.nan, 12.0, np.nan]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.6615242347139803)
      y = [np.nan, 4.0, 7.0, np.nan, np.nan]
      tau, p_value = stats.weightedtau(x, y)
      assert_approx_equal(tau, -0.6615242347139803)
  
  
  def test_segfault_issue_9710():
      # https://github.com/scipy/scipy/issues/9710
      # This test was created to check segfault
      # In issue SEGFAULT only repros in optimized builds after calling the function twice
      stats.weightedtau([1], [1.0])
      stats.weightedtau([1], [1.0])
      # The code below also caused SEGFAULT
      stats.weightedtau([np.nan], [52])
  
  
  def test_kendall_tau_large():
      n = 172
      # Test omit policy
      x = np.arange(n + 1).astype(float)
      y = np.arange(n + 1).astype(float)
      y[-1] = np.nan
      _, pval = stats.kendalltau(x, y, method='exact', nan_policy='omit')
      assert_equal(pval, 0.0)
  
  
  def test_weightedtau_vs_quadratic():
      # Trivial quadratic implementation, all parameters mandatory
      def wkq(x, y, rank, weigher, add):
          tot = conc = disc = u = v = 0
          for (i, j) in product(range(len(x)), range(len(x))):
              w = weigher(rank[i]) + weigher(rank[j]) if add \
                  else weigher(rank[i]) * weigher(rank[j])
              tot += w
              if x[i] == x[j]:
                  u += w
              if y[i] == y[j]:
                  v += w
              if x[i] < x[j] and y[i] < y[j] or x[i] > x[j] and y[i] > y[j]:
                  conc += w
              elif x[i] < x[j] and y[i] > y[j] or x[i] > x[j] and y[i] < y[j]:
                  disc += w
          return (conc - disc) / np.sqrt(tot - u) / np.sqrt(tot - v)
  
      np.random.seed(42)
      for s in range(3,10):
          a = []
          # Generate rankings with ties
          for i in range(s):
              a += [i]*i
          b = list(a)
          np.random.shuffle(a)
          np.random.shuffle(b)
          # First pass: use element indices as ranks
          rank = np.arange(len(a), dtype=np.intp)
          for _ in range(2):
              for add in [True, False]:
                  expected = wkq(a, b, rank, lambda x: 1./(x+1), add)
                  actual = stats.weightedtau(a, b, rank, lambda x: 1./(x+1), add).correlation
                  assert_approx_equal(expected, actual)
              # Second pass: use a random rank
              np.random.shuffle(rank)
  
  
  class TestFindRepeats:
  
      def test_basic(self):
          a = [1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 5]
          res, nums = stats.find_repeats(a)
          assert_array_equal(res, [1, 2, 3, 4])
          assert_array_equal(nums, [3, 3, 2, 2])
  
      def test_empty_result(self):
          # Check that empty arrays are returned when there are no repeats.
          for a in [[10, 20, 50, 30, 40], []]:
              repeated, counts = stats.find_repeats(a)
              assert_array_equal(repeated, [])
              assert_array_equal(counts, [])
  
  
  class TestRegression:
  
      def test_linregressBIGX(self):
          # W.II.F.  Regress BIG on X.
          result = stats.linregress(X, BIG)
          assert_almost_equal(result.intercept, 99999990)
          assert_almost_equal(result.rvalue, 1.0)
          # The uncertainty ought to be almost zero
          # since all points lie on a line
          assert_almost_equal(result.stderr, 0.0)
          assert_almost_equal(result.intercept_stderr, 0.0)
  
      def test_regressXX(self):
          # W.IV.B.  Regress X on X.
          # The constant should be exactly 0 and the regression coefficient
          # should be 1.  This is a perfectly valid regression and the
          # program should not complain.
          result = stats.linregress(X, X)
          assert_almost_equal(result.intercept, 0.0)
          assert_almost_equal(result.rvalue, 1.0)
          # The uncertainly on regression through two points ought to be 0
          assert_almost_equal(result.stderr, 0.0)
          assert_almost_equal(result.intercept_stderr, 0.0)
  
          # W.IV.C. Regress X on BIG and LITTLE (two predictors).  The program
          # should tell you that this model is "singular" because BIG and
          # LITTLE are linear combinations of each other.  Cryptic error
          # messages are unacceptable here.  Singularity is the most
          # fundamental regression error.
          #
          # Need to figure out how to handle multiple linear regression.
          # This is not obvious
  
      def test_regressZEROX(self):
          # W.IV.D. Regress ZERO on X.
          # The program should inform you that ZERO has no variance or it should
          # go ahead and compute the regression and report a correlation and
          # total sum of squares of exactly 0.
          result = stats.linregress(X, ZERO)
          assert_almost_equal(result.intercept, 0.0)
          assert_almost_equal(result.rvalue, 0.0)
  
      def test_regress_simple(self):
          # Regress a line with sinusoidal noise.
          x = np.linspace(0, 100, 100)
          y = 0.2 * np.linspace(0, 100, 100) + 10
          y += np.sin(np.linspace(0, 20, 100))
  
          result = stats.linregress(x, y)
          lr = stats._stats_mstats_common.LinregressResult
          assert_(isinstance(result, lr))
          assert_almost_equal(result.stderr, 2.3957814497838803e-3)
  
      def test_regress_alternative(self):
          # test alternative parameter
          x = np.linspace(0, 100, 100)
          y = 0.2 * np.linspace(0, 100, 100) + 10  # slope is greater than zero
          y += np.sin(np.linspace(0, 20, 100))
  
          with pytest.raises(ValueError, match="alternative must be 'less'..."):
              stats.linregress(x, y, alternative="ekki-ekki")
  
          res1 = stats.linregress(x, y, alternative="two-sided")
  
          # slope is greater than zero, so "less" p-value should be large
          res2 = stats.linregress(x, y, alternative="less")
          assert_allclose(res2.pvalue, 1 - (res1.pvalue / 2))
  
          # slope is greater than zero, so "greater" p-value should be small
          res3 = stats.linregress(x, y, alternative="greater")
          assert_allclose(res3.pvalue, res1.pvalue / 2)
  
          assert res1.rvalue == res2.rvalue == res3.rvalue
  
      def test_regress_against_R(self):
          # test against R `lm`
          # options(digits=16)
          # x <- c(151, 174, 138, 186, 128, 136, 179, 163, 152, 131)
          # y <- c(63, 81, 56, 91, 47, 57, 76, 72, 62, 48)
          # relation <- lm(y~x)
          # print(summary(relation))
  
          x = [151, 174, 138, 186, 128, 136, 179, 163, 152, 131]
          y = [63, 81, 56, 91, 47, 57, 76, 72, 62, 48]
          res = stats.linregress(x, y, alternative="two-sided")
          # expected values from R's `lm` above
          assert_allclose(res.slope, 0.6746104491292)
          assert_allclose(res.intercept, -38.4550870760770)
          assert_allclose(res.rvalue, np.sqrt(0.95478224775))
          assert_allclose(res.pvalue, 1.16440531074e-06)
          assert_allclose(res.stderr, 0.0519051424731)
          assert_allclose(res.intercept_stderr, 8.0490133029927)
  
      def test_regress_simple_onearg_rows(self):
          # Regress a line w sinusoidal noise,
          # with a single input of shape (2, N)
          x = np.linspace(0, 100, 100)
          y = 0.2 * np.linspace(0, 100, 100) + 10
          y += np.sin(np.linspace(0, 20, 100))
          rows = np.vstack((x, y))
  
          result = stats.linregress(rows)
          assert_almost_equal(result.stderr, 2.3957814497838803e-3)
          assert_almost_equal(result.intercept_stderr, 1.3866936078570702e-1)
  
      def test_regress_simple_onearg_cols(self):
          x = np.linspace(0, 100, 100)
          y = 0.2 * np.linspace(0, 100, 100) + 10
          y += np.sin(np.linspace(0, 20, 100))
          columns = np.hstack((np.expand_dims(x, 1), np.expand_dims(y, 1)))
  
          result = stats.linregress(columns)
          assert_almost_equal(result.stderr, 2.3957814497838803e-3)
          assert_almost_equal(result.intercept_stderr, 1.3866936078570702e-1)
  
      def test_regress_shape_error(self):
          # Check that a single input argument to linregress with wrong shape
          # results in a ValueError.
          assert_raises(ValueError, stats.linregress, np.ones((3, 3)))
  
      def test_linregress(self):
          # compared with multivariate ols with pinv
          x = np.arange(11)
          y = np.arange(5, 16)
          y[[(1), (-2)]] -= 1
          y[[(0), (-1)]] += 1
  
          result = stats.linregress(x, y)
  
          # This test used to use 'assert_array_almost_equal' but its
          # formualtion got confusing since LinregressResult became
          # _lib._bunch._make_tuple_bunch instead of namedtuple
          # (for backwards compatibility, see PR #12983)
          assert_ae = lambda x, y: assert_almost_equal(x, y, decimal=14)
          assert_ae(result.slope, 1.0)
          assert_ae(result.intercept, 5.0)
          assert_ae(result.rvalue, 0.98229948625750)
          assert_ae(result.pvalue, 7.45259691e-008)
          assert_ae(result.stderr, 0.063564172616372733)
          assert_ae(result.intercept_stderr, 0.37605071654517686)
  
      def test_regress_simple_negative_cor(self):
          # If the slope of the regression is negative the factor R tend
          # to -1 not 1.  Sometimes rounding errors makes it < -1
          # leading to stderr being NaN.
          a, n = 1e-71, 100000
          x = np.linspace(a, 2 * a, n)
          y = np.linspace(2 * a, a, n)
          result = stats.linregress(x, y)
  
          # Make sure propagated numerical errors
          # did not bring rvalue below -1 (or were coersced)
          assert_(result.rvalue >= -1)
          assert_almost_equal(result.rvalue, -1)
  
          # slope and intercept stderror should stay numeric
          assert_(not np.isnan(result.stderr))
          assert_(not np.isnan(result.intercept_stderr))
  
      def test_linregress_result_attributes(self):
          x = np.linspace(0, 100, 100)
          y = 0.2 * np.linspace(0, 100, 100) + 10
          y += np.sin(np.linspace(0, 20, 100))
          result = stats.linregress(x, y)
  
          # Result is of a correct class
          lr = stats._stats_mstats_common.LinregressResult
          assert_(isinstance(result, lr))
  
          # LinregressResult elements have correct names
          attributes = ('slope', 'intercept', 'rvalue', 'pvalue', 'stderr')
          check_named_results(result, attributes)
          # Also check that the extra attribute (intercept_stderr) is present
          assert 'intercept_stderr' in dir(result)
  
      def test_regress_two_inputs(self):
          # Regress a simple line formed by two points.
          x = np.arange(2)
          y = np.arange(3, 5)
          result = stats.linregress(x, y)
  
          # Non-horizontal line
          assert_almost_equal(result.pvalue, 0.0)
  
          # Zero error through two points
          assert_almost_equal(result.stderr, 0.0)
          assert_almost_equal(result.intercept_stderr, 0.0)
  
      def test_regress_two_inputs_horizontal_line(self):
          # Regress a horizontal line formed by two points.
          x = np.arange(2)
          y = np.ones(2)
          result = stats.linregress(x, y)
  
          # Horizontal line
          assert_almost_equal(result.pvalue, 1.0)
  
          # Zero error through two points
          assert_almost_equal(result.stderr, 0.0)
          assert_almost_equal(result.intercept_stderr, 0.0)
  
      def test_nist_norris(self):
          x = [0.2, 337.4, 118.2, 884.6, 10.1, 226.5, 666.3, 996.3, 448.6, 777.0,
               558.2, 0.4, 0.6, 775.5, 666.9, 338.0, 447.5, 11.6, 556.0, 228.1,
               995.8, 887.6, 120.2, 0.3, 0.3, 556.8, 339.1, 887.2, 999.0, 779.0,
               11.1, 118.3, 229.2, 669.1, 448.9, 0.5]
  
          y = [0.1, 338.8, 118.1, 888.0, 9.2, 228.1, 668.5, 998.5, 449.1, 778.9,
               559.2, 0.3, 0.1, 778.1, 668.8, 339.3, 448.9, 10.8, 557.7, 228.3,
               998.0, 888.8, 119.6, 0.3, 0.6, 557.6, 339.3, 888.0, 998.5, 778.9,
               10.2, 117.6, 228.9, 668.4, 449.2, 0.2]
  
          result = stats.linregress(x, y)
  
          assert_almost_equal(result.slope, 1.00211681802045)
          assert_almost_equal(result.intercept, -0.262323073774029)
          assert_almost_equal(result.rvalue**2, 0.999993745883712)
          assert_almost_equal(result.pvalue, 0.0)
          assert_almost_equal(result.stderr, 0.00042979684820)
          assert_almost_equal(result.intercept_stderr, 0.23281823430153)
  
      def test_compare_to_polyfit(self):
          x = np.linspace(0, 100, 100)
          y = 0.2 * np.linspace(0, 100, 100) + 10
          y += np.sin(np.linspace(0, 20, 100))
          result = stats.linregress(x, y)
          poly = np.polyfit(x, y, 1)  # Fit 1st degree polynomial
  
          # Make sure linear regression slope and intercept
          # match with results from numpy polyfit
          assert_almost_equal(result.slope, poly[0])
          assert_almost_equal(result.intercept, poly[1])
  
      def test_empty_input(self):
          assert_raises(ValueError, stats.linregress, [], [])
  
      def test_nan_input(self):
          x = np.arange(10.)
          x[9] = np.nan
  
          with np.errstate(invalid="ignore"):
              result = stats.linregress(x, x)
  
          # Make sure the resut still comes back as `LinregressResult`
          lr = stats._stats_mstats_common.LinregressResult
          assert_(isinstance(result, lr))
          assert_array_equal(result, (np.nan,)*5)
          assert_equal(result.intercept_stderr, np.nan)
  
      def test_identical_x(self):
          x = np.zeros(10)
          y = np.random.random(10)
          msg = "Cannot calculate a linear regression"
          with assert_raises(ValueError, match=msg):
              stats.linregress(x, y)
  
  def test_theilslopes():
      # Basic slope test.
      slope, intercept, lower, upper = stats.theilslopes([0,1,1])
      assert_almost_equal(slope, 0.5)
      assert_almost_equal(intercept, 0.5)
  
      msg = ("method must be either 'joint' or 'separate'."
             "'joint_separate' is invalid.")
      with pytest.raises(ValueError, match=msg):
          stats.theilslopes([0, 1, 1], method='joint_separate')
  
      slope, intercept, lower, upper = stats.theilslopes([0, 1, 1],
                                                         method='joint')
      assert_almost_equal(slope, 0.5)
      assert_almost_equal(intercept, 0.0)
  
      # Test of confidence intervals.
      x = [1, 2, 3, 4, 10, 12, 18]
      y = [9, 15, 19, 20, 45, 55, 78]
      slope, intercept, lower, upper = stats.theilslopes(y, x, 0.07,
                                                         method='separate')
      assert_almost_equal(slope, 4)
      assert_almost_equal(intercept, 4.0)
      assert_almost_equal(upper, 4.38, decimal=2)
      assert_almost_equal(lower, 3.71, decimal=2)
  
      slope, intercept, lower, upper = stats.theilslopes(y, x, 0.07,
                                                         method='joint')
      assert_almost_equal(slope, 4)
      assert_almost_equal(intercept, 6.0)
      assert_almost_equal(upper, 4.38, decimal=2)
      assert_almost_equal(lower, 3.71, decimal=2)
  
  
  def test_cumfreq():
      x = [1, 4, 2, 1, 3, 1]
      cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4)
      assert_array_almost_equal(cumfreqs, np.array([3., 4., 5., 6.]))
      cumfreqs, lowlim, binsize, extrapoints = stats.cumfreq(x, numbins=4,
                                                        defaultreallimits=(1.5, 5))
      assert_(extrapoints == 3)
  
      # test for namedtuple attribute results
      attributes = ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')
      res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
      check_named_results(res, attributes)
  
  
  def test_relfreq():
      a = np.array([1, 4, 2, 1, 3, 1])
      relfreqs, lowlim, binsize, extrapoints = stats.relfreq(a, numbins=4)
      assert_array_almost_equal(relfreqs,
                                array([0.5, 0.16666667, 0.16666667, 0.16666667]))
  
      # test for namedtuple attribute results
      attributes = ('frequency', 'lowerlimit', 'binsize', 'extrapoints')
      res = stats.relfreq(a, numbins=4)
      check_named_results(res, attributes)
  
      # check array_like input is accepted
      relfreqs2, lowlim, binsize, extrapoints = stats.relfreq([1, 4, 2, 1, 3, 1],
                                                              numbins=4)
      assert_array_almost_equal(relfreqs, relfreqs2)
  
  
  class TestScoreatpercentile:
      def setup_method(self):
          self.a1 = [3, 4, 5, 10, -3, -5, 6]
          self.a2 = [3, -6, -2, 8, 7, 4, 2, 1]
          self.a3 = [3., 4, 5, 10, -3, -5, -6, 7.0]
  
      def test_basic(self):
          x = arange(8) * 0.5
          assert_equal(stats.scoreatpercentile(x, 0), 0.)
          assert_equal(stats.scoreatpercentile(x, 100), 3.5)
          assert_equal(stats.scoreatpercentile(x, 50), 1.75)
  
      def test_fraction(self):
          scoreatperc = stats.scoreatpercentile
  
          # Test defaults
          assert_equal(scoreatperc(list(range(10)), 50), 4.5)
          assert_equal(scoreatperc(list(range(10)), 50, (2,7)), 4.5)
          assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8)), 4.5)
          assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10,100)), 55)
          assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10)), 5.5)
  
          # explicitly specify interpolation_method 'fraction' (the default)
          assert_equal(scoreatperc(list(range(10)), 50, interpolation_method='fraction'),
                       4.5)
          assert_equal(scoreatperc(list(range(10)), 50, limit=(2, 7),
                                   interpolation_method='fraction'),
                       4.5)
          assert_equal(scoreatperc(list(range(100)), 50, limit=(1, 8),
                                   interpolation_method='fraction'),
                       4.5)
          assert_equal(scoreatperc(np.array([1, 10,100]), 50, (10, 100),
                                   interpolation_method='fraction'),
                       55)
          assert_equal(scoreatperc(np.array([1, 10,100]), 50, (1,10),
                                   interpolation_method='fraction'),
                       5.5)
  
      def test_lower_higher(self):
          scoreatperc = stats.scoreatpercentile
  
          # interpolation_method 'lower'/'higher'
          assert_equal(scoreatperc(list(range(10)), 50,
                                   interpolation_method='lower'), 4)
          assert_equal(scoreatperc(list(range(10)), 50,
                                   interpolation_method='higher'), 5)
          assert_equal(scoreatperc(list(range(10)), 50, (2,7),
                                   interpolation_method='lower'), 4)
          assert_equal(scoreatperc(list(range(10)), 50, limit=(2,7),
                                   interpolation_method='higher'), 5)
          assert_equal(scoreatperc(list(range(100)), 50, (1,8),
                                   interpolation_method='lower'), 4)
          assert_equal(scoreatperc(list(range(100)), 50, (1,8),
                                   interpolation_method='higher'), 5)
          assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (10, 100),
                                   interpolation_method='lower'), 10)
          assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(10, 100),
                                   interpolation_method='higher'), 100)
          assert_equal(scoreatperc(np.array([1, 10, 100]), 50, (1, 10),
                                   interpolation_method='lower'), 1)
          assert_equal(scoreatperc(np.array([1, 10, 100]), 50, limit=(1, 10),
                                   interpolation_method='higher'), 10)
  
      def test_sequence_per(self):
          x = arange(8) * 0.5
          expected = np.array([0, 3.5, 1.75])
          res = stats.scoreatpercentile(x, [0, 100, 50])
          assert_allclose(res, expected)
          assert_(isinstance(res, np.ndarray))
          # Test with ndarray.  Regression test for gh-2861
          assert_allclose(stats.scoreatpercentile(x, np.array([0, 100, 50])),
                          expected)
          # Also test combination of 2-D array, axis not None and array-like per
          res2 = stats.scoreatpercentile(np.arange(12).reshape((3,4)),
                                         np.array([0, 1, 100, 100]), axis=1)
          expected2 = array([[0, 4, 8],
                             [0.03, 4.03, 8.03],
                             [3, 7, 11],
                             [3, 7, 11]])
          assert_allclose(res2, expected2)
  
      def test_axis(self):
          scoreatperc = stats.scoreatpercentile
          x = arange(12).reshape(3, 4)
  
          assert_equal(scoreatperc(x, (25, 50, 100)), [2.75, 5.5, 11.0])
  
          r0 = [[2, 3, 4, 5], [4, 5, 6, 7], [8, 9, 10, 11]]
          assert_equal(scoreatperc(x, (25, 50, 100), axis=0), r0)
  
          r1 = [[0.75, 4.75, 8.75], [1.5, 5.5, 9.5], [3, 7, 11]]
          assert_equal(scoreatperc(x, (25, 50, 100), axis=1), r1)
  
          x = array([[1, 1, 1],
                     [1, 1, 1],
                     [4, 4, 3],
                     [1, 1, 1],
                     [1, 1, 1]])
          score = stats.scoreatpercentile(x, 50)
          assert_equal(score.shape, ())
          assert_equal(score, 1.0)
          score = stats.scoreatpercentile(x, 50, axis=0)
          assert_equal(score.shape, (3,))
          assert_equal(score, [1, 1, 1])
  
      def test_exception(self):
          assert_raises(ValueError, stats.scoreatpercentile, [1, 2], 56,
              interpolation_method='foobar')
          assert_raises(ValueError, stats.scoreatpercentile, [1], 101)
          assert_raises(ValueError, stats.scoreatpercentile, [1], -1)
  
      def test_empty(self):
          assert_equal(stats.scoreatpercentile([], 50), np.nan)
          assert_equal(stats.scoreatpercentile(np.array([[], []]), 50), np.nan)
          assert_equal(stats.scoreatpercentile([], [50, 99]), [np.nan, np.nan])
  
  
  class TestMode:
  
      deprecation_msg = r"Support for non-numeric arrays has been deprecated"
  
      def test_empty(self):
          vals, counts = stats.mode([])
          assert_equal(vals, np.array([]))
          assert_equal(counts, np.array([]))
  
      def test_scalar(self):
          vals, counts = stats.mode(4.)
          assert_equal(vals, np.array([4.]))
          assert_equal(counts, np.array([1]))
  
      def test_basic(self):
          data1 = [3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
          vals = stats.mode(data1)
          assert_equal(vals[0], 6)
          assert_equal(vals[1], 3)
  
      def test_axes(self):
          data1 = [10, 10, 30, 40]
          data2 = [10, 10, 10, 10]
          data3 = [20, 10, 20, 20]
          data4 = [30, 30, 30, 30]
          data5 = [40, 30, 30, 30]
          arr = np.array([data1, data2, data3, data4, data5])
  
          vals = stats.mode(arr, axis=None)
          assert_equal(vals[0], np.array(30))
          assert_equal(vals[1], np.array(8))
  
          vals = stats.mode(arr, axis=0)
          assert_equal(vals[0], np.array([10, 10, 30, 30]))
          assert_equal(vals[1], np.array([2, 3, 3, 2]))
  
          vals = stats.mode(arr, axis=1)
          assert_equal(vals[0], np.array([10, 10, 20, 30, 30]))
          assert_equal(vals[1], np.array([2, 4, 3, 4, 3]))
  
      @pytest.mark.parametrize('axis', np.arange(-4, 0))
      def test_negative_axes_gh_15375(self, axis):
          np.random.seed(984213899)
          a = np.random.rand(10, 11, 12, 13)
          res0 = stats.mode(a, axis=a.ndim+axis)
          res1 = stats.mode(a, axis=axis)
          np.testing.assert_array_equal(res0, res1)
  
      def test_strings(self):
          data1 = ['rain', 'showers', 'showers']
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              vals = stats.mode(data1)
          assert_equal(vals[0], 'showers')
          assert_equal(vals[1], 2)
  
      def test_mixed_objects(self):
          objects = [10, True, np.nan, 'hello', 10]
          arr = np.empty((5,), dtype=object)
          arr[:] = objects
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              vals = stats.mode(arr)
          assert_equal(vals[0], 10)
          assert_equal(vals[1], 2)
  
      def test_objects(self):
          # Python objects must be sortable (le + eq) and have ne defined
          # for np.unique to work. hash is for set.
          class Point:
              def __init__(self, x):
                  self.x = x
  
              def __eq__(self, other):
                  return self.x == other.x
  
              def __ne__(self, other):
                  return self.x != other.x
  
              def __lt__(self, other):
                  return self.x < other.x
  
              def __hash__(self):
                  return hash(self.x)
  
          points = [Point(x) for x in [1, 2, 3, 4, 3, 2, 2, 2]]
          arr = np.empty((8,), dtype=object)
          arr[:] = points
          assert_(len(set(points)) == 4)
          assert_equal(np.unique(arr).shape, (4,))
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              vals = stats.mode(arr)
  
          assert_equal(vals[0], Point(2))
          assert_equal(vals[1], 4)
  
      def test_mode_result_attributes(self):
          data1 = [3, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
          data2 = []
          actual = stats.mode(data1)
          attributes = ('mode', 'count')
          check_named_results(actual, attributes)
          actual2 = stats.mode(data2)
          check_named_results(actual2, attributes)
  
      def test_mode_nan(self):
          data1 = [3, np.nan, 5, 1, 10, 23, 3, 2, 6, 8, 6, 10, 6]
          actual = stats.mode(data1)
          assert_equal(actual, (6, 3))
  
          actual = stats.mode(data1, nan_policy='omit')
          assert_equal(actual, (6, 3))
          assert_raises(ValueError, stats.mode, data1, nan_policy='raise')
          assert_raises(ValueError, stats.mode, data1, nan_policy='foobar')
  
      @pytest.mark.parametrize("data", [
          [3, 5, 1, 1, 3],
          [3, np.nan, 5, 1, 1, 3],
          [3, 5, 1],
          [3, np.nan, 5, 1],
      ])
      def test_smallest_equal(self, data):
          result = stats.mode(data, nan_policy='omit')
          assert_equal(result[0], 1)
  
      def test_obj_arrays_ndim(self):
          # regression test for gh-9645: `mode` fails for object arrays w/ndim > 1
          data = [['Oxidation'], ['Oxidation'], ['Polymerization'], ['Reduction']]
          ar = np.array(data, dtype=object)
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              m = stats.mode(ar, axis=0)
          assert np.all(m.mode == 'Oxidation') and m.mode.shape == (1,)
          assert np.all(m.count == 2) and m.count.shape == (1,)
  
          data1 = data + [[np.nan]]
          ar1 = np.array(data1, dtype=object)
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              m = stats.mode(ar1, axis=0)
          assert np.all(m.mode == 'Oxidation') and m.mode.shape == (1,)
          assert np.all(m.count == 2) and m.count.shape == (1,)
  
      @pytest.mark.parametrize('axis', np.arange(-3, 3))
      @pytest.mark.parametrize('dtype', [np.float64, 'object'])
      def test_mode_shape_gh_9955(self, axis, dtype):
          rng = np.random.default_rng(984213899)
          a = rng.uniform(size=(3, 4, 5)).astype(dtype)
          if dtype == 'object':
              with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
                  res = stats.mode(a, axis=axis)
          else:
              res = stats.mode(a, axis=axis)
          reference_shape = list(a.shape)
          reference_shape.pop(axis)
          np.testing.assert_array_equal(res.mode.shape, reference_shape)
          np.testing.assert_array_equal(res.count.shape, reference_shape)
  
      def test_nan_policy_propagate_gh_9815(self):
          # mode should treat np.nan as it would any other object when
          # nan_policy='propagate'
          a = [2, np.nan, 1, np.nan]
          if NumpyVersion(np.__version__) >= '1.21.0':
              res = stats.mode(a)
              assert np.isnan(res.mode) and res.count == 2
  
          # mode should work on object arrays. There were issues when
          # objects do not support comparison operations.
          a = np.array(a, dtype='object')
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              res = stats.mode(a)
          assert np.isnan(res.mode) and res.count == 2
  
          a = np.array([10, True, 'hello', 10], dtype='object')
          with pytest.warns(DeprecationWarning, match=self.deprecation_msg):
              res = stats.mode(a)
          assert_array_equal(res, (10, 2))
  
  
  class TestSEM:
  
      testcase = [1, 2, 3, 4]
      scalar_testcase = 4.
  
      def test_sem(self):
          # This is not in R, so used:
          #     sqrt(var(testcase)*3/4)/sqrt(3)
  
          # y = stats.sem(self.shoes[0])
          # assert_approx_equal(y,0.775177399)
          with suppress_warnings() as sup, np.errstate(invalid="ignore"):
              sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
              y = stats.sem(self.scalar_testcase)
          assert_(np.isnan(y))
  
          y = stats.sem(self.testcase)
          assert_approx_equal(y, 0.6454972244)
          n = len(self.testcase)
          assert_allclose(stats.sem(self.testcase, ddof=0) * np.sqrt(n/(n-2)),
                          stats.sem(self.testcase, ddof=2))
  
          x = np.arange(10.)
          x[9] = np.nan
          assert_equal(stats.sem(x), np.nan)
          assert_equal(stats.sem(x, nan_policy='omit'), 0.9128709291752769)
          assert_raises(ValueError, stats.sem, x, nan_policy='raise')
          assert_raises(ValueError, stats.sem, x, nan_policy='foobar')
  
  
  class TestZmapZscore:
  
      @pytest.mark.parametrize(
          'x, y',
          [([1, 2, 3, 4], [1, 2, 3, 4]),
           ([1, 2, 3], [0, 1, 2, 3, 4])]
      )
      def test_zmap(self, x, y):
          z = stats.zmap(x, y)
          # For these simple cases, calculate the expected result directly
          # by using the formula for the z-score.
          expected = (x - np.mean(y))/np.std(y)
          assert_allclose(z, expected, rtol=1e-12)
  
      def test_zmap_axis(self):
          # Test use of 'axis' keyword in zmap.
          x = np.array([[0.0, 0.0, 1.0, 1.0],
                        [1.0, 1.0, 1.0, 2.0],
                        [2.0, 0.0, 2.0, 0.0]])
  
          t1 = 1.0/np.sqrt(2.0/3)
          t2 = np.sqrt(3.)/3
          t3 = np.sqrt(2.)
  
          z0 = stats.zmap(x, x, axis=0)
          z1 = stats.zmap(x, x, axis=1)
  
          z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
                         [0.0, t3, -t3/2, t1],
                         [t1, -t3/2, t3, -t1]]
          z1_expected = [[-1.0, -1.0, 1.0, 1.0],
                         [-t2, -t2, -t2, np.sqrt(3.)],
                         [1.0, -1.0, 1.0, -1.0]]
  
          assert_array_almost_equal(z0, z0_expected)
          assert_array_almost_equal(z1, z1_expected)
  
      def test_zmap_ddof(self):
          # Test use of 'ddof' keyword in zmap.
          x = np.array([[0.0, 0.0, 1.0, 1.0],
                        [0.0, 1.0, 2.0, 3.0]])
  
          z = stats.zmap(x, x, axis=1, ddof=1)
  
          z0_expected = np.array([-0.5, -0.5, 0.5, 0.5])/(1.0/np.sqrt(3))
          z1_expected = np.array([-1.5, -0.5, 0.5, 1.5])/(np.sqrt(5./3))
          assert_array_almost_equal(z[0], z0_expected)
          assert_array_almost_equal(z[1], z1_expected)
  
      @pytest.mark.parametrize('ddof', [0, 2])
      def test_zmap_nan_policy_omit(self, ddof):
          # nans in `scores` are propagated, regardless of `nan_policy`.
          # `nan_policy` only affects how nans in `compare` are handled.
          scores = np.array([-3, -1, 2, np.nan])
          compare = np.array([-8, -3, 2, 7, 12, np.nan])
          z = stats.zmap(scores, compare, ddof=ddof, nan_policy='omit')
          assert_allclose(z, stats.zmap(scores, compare[~np.isnan(compare)],
                                        ddof=ddof))
  
      @pytest.mark.parametrize('ddof', [0, 2])
      def test_zmap_nan_policy_omit_with_axis(self, ddof):
          scores = np.arange(-5.0, 9.0).reshape(2, -1)
          compare = np.linspace(-8, 6, 24).reshape(2, -1)
          compare[0, 4] = np.nan
          compare[0, 6] = np.nan
          compare[1, 1] = np.nan
          z = stats.zmap(scores, compare, nan_policy='omit', axis=1, ddof=ddof)
          expected = np.array([stats.zmap(scores[0],
                                          compare[0][~np.isnan(compare[0])],
                                          ddof=ddof),
                               stats.zmap(scores[1],
                                          compare[1][~np.isnan(compare[1])],
                                          ddof=ddof)])
          assert_allclose(z, expected, rtol=1e-14)
  
      def test_zmap_nan_policy_raise(self):
          scores = np.array([1, 2, 3])
          compare = np.array([-8, -3, 2, 7, 12, np.nan])
          with pytest.raises(ValueError, match='input contains nan'):
              stats.zmap(scores, compare, nan_policy='raise')
  
      def test_zscore(self):
          # not in R, so tested by using:
          #    (testcase[i] - mean(testcase, axis=0)) / sqrt(var(testcase) * 3/4)
          y = stats.zscore([1, 2, 3, 4])
          desired = ([-1.3416407864999, -0.44721359549996, 0.44721359549996,
                      1.3416407864999])
          assert_array_almost_equal(desired, y, decimal=12)
  
      def test_zscore_axis(self):
          # Test use of 'axis' keyword in zscore.
          x = np.array([[0.0, 0.0, 1.0, 1.0],
                        [1.0, 1.0, 1.0, 2.0],
                        [2.0, 0.0, 2.0, 0.0]])
  
          t1 = 1.0/np.sqrt(2.0/3)
          t2 = np.sqrt(3.)/3
          t3 = np.sqrt(2.)
  
          z0 = stats.zscore(x, axis=0)
          z1 = stats.zscore(x, axis=1)
  
          z0_expected = [[-t1, -t3/2, -t3/2, 0.0],
                         [0.0, t3, -t3/2, t1],
                         [t1, -t3/2, t3, -t1]]
          z1_expected = [[-1.0, -1.0, 1.0, 1.0],
                         [-t2, -t2, -t2, np.sqrt(3.)],
                         [1.0, -1.0, 1.0, -1.0]]
  
          assert_array_almost_equal(z0, z0_expected)
          assert_array_almost_equal(z1, z1_expected)
  
      def test_zscore_ddof(self):
          # Test use of 'ddof' keyword in zscore.
          x = np.array([[0.0, 0.0, 1.0, 1.0],
                        [0.0, 1.0, 2.0, 3.0]])
  
          z = stats.zscore(x, axis=1, ddof=1)
  
          z0_expected = np.array([-0.5, -0.5, 0.5, 0.5])/(1.0/np.sqrt(3))
          z1_expected = np.array([-1.5, -0.5, 0.5, 1.5])/(np.sqrt(5./3))
          assert_array_almost_equal(z[0], z0_expected)
          assert_array_almost_equal(z[1], z1_expected)
  
      def test_zscore_nan_propagate(self):
          x = np.array([1, 2, np.nan, 4, 5])
          z = stats.zscore(x, nan_policy='propagate')
          assert all(np.isnan(z))
  
      def test_zscore_nan_omit(self):
          x = np.array([1, 2, np.nan, 4, 5])
  
          z = stats.zscore(x, nan_policy='omit')
  
          expected = np.array([-1.2649110640673518,
                               -0.6324555320336759,
                               np.nan,
                               0.6324555320336759,
                               1.2649110640673518
                               ])
          assert_array_almost_equal(z, expected)
  
      def test_zscore_nan_omit_with_ddof(self):
          x = np.array([np.nan, 1.0, 3.0, 5.0, 7.0, 9.0])
          z = stats.zscore(x, ddof=1, nan_policy='omit')
          expected = np.r_[np.nan, stats.zscore(x[1:], ddof=1)]
          assert_allclose(z, expected, rtol=1e-13)
  
      def test_zscore_nan_raise(self):
          x = np.array([1, 2, np.nan, 4, 5])
  
          assert_raises(ValueError, stats.zscore, x, nan_policy='raise')
  
      def test_zscore_constant_input_1d(self):
          x = [-0.087] * 3
          z = stats.zscore(x)
          assert_equal(z, np.full(len(x), np.nan))
  
      def test_zscore_constant_input_2d(self):
          x = np.array([[10.0, 10.0, 10.0, 10.0],
                        [10.0, 11.0, 12.0, 13.0]])
          z0 = stats.zscore(x, axis=0)
          assert_equal(z0, np.array([[np.nan, -1.0, -1.0, -1.0],
                                     [np.nan, 1.0, 1.0, 1.0]]))
          z1 = stats.zscore(x, axis=1)
          assert_equal(z1, np.array([[np.nan, np.nan, np.nan, np.nan],
                                     stats.zscore(x[1])]))
          z = stats.zscore(x, axis=None)
          assert_equal(z, stats.zscore(x.ravel()).reshape(x.shape))
  
          y = np.ones((3, 6))
          z = stats.zscore(y, axis=None)
          assert_equal(z, np.full(y.shape, np.nan))
  
      def test_zscore_constant_input_2d_nan_policy_omit(self):
          x = np.array([[10.0, 10.0, 10.0, 10.0],
                        [10.0, 11.0, 12.0, np.nan],
                        [10.0, 12.0, np.nan, 10.0]])
          z0 = stats.zscore(x, nan_policy='omit', axis=0)
          s = np.sqrt(3/2)
          s2 = np.sqrt(2)
          assert_allclose(z0, np.array([[np.nan, -s, -1.0, np.nan],
                                        [np.nan, 0, 1.0, np.nan],
                                        [np.nan, s, np.nan, np.nan]]))
          z1 = stats.zscore(x, nan_policy='omit', axis=1)
          assert_allclose(z1, np.array([[np.nan, np.nan, np.nan, np.nan],
                                        [-s, 0, s, np.nan],
                                        [-s2/2, s2, np.nan, -s2/2]]))
  
      def test_zscore_2d_all_nan_row(self):
          # A row is all nan, and we use axis=1.
          x = np.array([[np.nan, np.nan, np.nan, np.nan],
                        [10.0, 10.0, 12.0, 12.0]])
          z = stats.zscore(x, nan_policy='omit', axis=1)
          assert_equal(z, np.array([[np.nan, np.nan, np.nan, np.nan],
                                    [-1.0, -1.0, 1.0, 1.0]]))
  
      def test_zscore_2d_all_nan(self):
          # The entire 2d array is nan, and we use axis=None.
          y = np.full((2, 3), np.nan)
          z = stats.zscore(y, nan_policy='omit', axis=None)
          assert_equal(z, y)
  
      @pytest.mark.parametrize('x', [np.array([]), np.zeros((3, 0, 5))])
      def test_zscore_empty_input(self, x):
          z = stats.zscore(x)
          assert_equal(z, x)
  
      def test_gzscore_normal_array(self):
          z = stats.gzscore([1, 2, 3, 4])
          desired = ([-1.526072095151, -0.194700599824, 0.584101799472,
                      1.136670895503])
          assert_allclose(desired, z)
  
      def test_gzscore_masked_array(self):
          x = np.array([1, 2, -1, 3, 4])
          mx = np.ma.masked_array(x, mask=[0, 0, 1, 0, 0])
          z = stats.gzscore(mx)
          desired = ([-1.526072095151, -0.194700599824, np.inf, 0.584101799472,
                      1.136670895503])
          assert_allclose(desired, z)
  
  
  class TestMedianAbsDeviation:
      def setup_class(self):
          self.dat_nan = np.array([2.20, 2.20, 2.4, 2.4, 2.5, 2.7, 2.8, 2.9,
                                   3.03, 3.03, 3.10, 3.37, 3.4, 3.4, 3.4, 3.5,
                                   3.6, 3.7, 3.7, 3.7, 3.7, 3.77, 5.28, np.nan])
          self.dat = np.array([2.20, 2.20, 2.4, 2.4, 2.5, 2.7, 2.8, 2.9, 3.03,
                               3.03, 3.10, 3.37, 3.4, 3.4, 3.4, 3.5, 3.6, 3.7,
                               3.7, 3.7, 3.7, 3.77, 5.28, 28.95])
  
      def test_median_abs_deviation(self):
          assert_almost_equal(stats.median_abs_deviation(self.dat, axis=None),
                              0.355)
          dat = self.dat.reshape(6, 4)
          mad = stats.median_abs_deviation(dat, axis=0)
          mad_expected = np.asarray([0.435, 0.5, 0.45, 0.4])
          assert_array_almost_equal(mad, mad_expected)
  
      def test_mad_nan_omit(self):
          mad = stats.median_abs_deviation(self.dat_nan, nan_policy='omit')
          assert_almost_equal(mad, 0.34)
  
      def test_axis_and_nan(self):
          x = np.array([[1.0, 2.0, 3.0, 4.0, np.nan],
                        [1.0, 4.0, 5.0, 8.0, 9.0]])
          mad = stats.median_abs_deviation(x, axis=1)
          assert_equal(mad, np.array([np.nan, 3.0]))
  
      def test_nan_policy_omit_with_inf(sef):
          z = np.array([1, 3, 4, 6, 99, np.nan, np.inf])
          mad = stats.median_abs_deviation(z, nan_policy='omit')
          assert_equal(mad, 3.0)
  
      @pytest.mark.parametrize('axis', [0, 1, 2, None])
      def test_size_zero_with_axis(self, axis):
          x = np.zeros((3, 0, 4))
          mad = stats.median_abs_deviation(x, axis=axis)
          assert_equal(mad, np.full_like(x.sum(axis=axis), fill_value=np.nan))
  
      @pytest.mark.parametrize('nan_policy, expected',
                               [('omit', np.array([np.nan, 1.5, 1.5])),
                                ('propagate', np.array([np.nan, np.nan, 1.5]))])
      def test_nan_policy_with_axis(self, nan_policy, expected):
          x = np.array([[np.nan, np.nan, np.nan, np.nan, np.nan, np.nan],
                        [1, 5, 3, 6, np.nan, np.nan],
                        [5, 6, 7, 9, 9, 10]])
          mad = stats.median_abs_deviation(x, nan_policy=nan_policy, axis=1)
          assert_equal(mad, expected)
  
      @pytest.mark.parametrize('axis, expected',
                               [(1, [2.5, 2.0, 12.0]), (None, 4.5)])
      def test_center_mean_with_nan(self, axis, expected):
          x = np.array([[1, 2, 4, 9, np.nan],
                        [0, 1, 1, 1, 12],
                        [-10, -10, -10, 20, 20]])
          mad = stats.median_abs_deviation(x, center=np.mean, nan_policy='omit',
                                           axis=axis)
          assert_allclose(mad, expected, rtol=1e-15, atol=1e-15)
  
      def test_center_not_callable(self):
          with pytest.raises(TypeError, match='callable'):
              stats.median_abs_deviation([1, 2, 3, 5], center=99)
  
  
  def _check_warnings(warn_list, expected_type, expected_len):
      """
      Checks that all of the warnings from a list returned by
      `warnings.catch_all(record=True)` are of the required type and that the list
      contains expected number of warnings.
      """
      assert_equal(len(warn_list), expected_len, "number of warnings")
      for warn_ in warn_list:
          assert_(warn_.category is expected_type)
  
  
  class TestIQR:
  
      def test_basic(self):
          x = np.arange(8) * 0.5
          np.random.shuffle(x)
          assert_equal(stats.iqr(x), 1.75)
  
      def test_api(self):
          d = np.ones((5, 5))
          stats.iqr(d)
          stats.iqr(d, None)
          stats.iqr(d, 1)
          stats.iqr(d, (0, 1))
          stats.iqr(d, None, (10, 90))
          stats.iqr(d, None, (30, 20), 1.0)
          stats.iqr(d, None, (25, 75), 1.5, 'propagate')
          stats.iqr(d, None, (50, 50), 'normal', 'raise', 'linear')
          stats.iqr(d, None, (25, 75), -0.4, 'omit', 'lower', True)
  
      def test_empty(self):
          assert_equal(stats.iqr([]), np.nan)
          assert_equal(stats.iqr(np.arange(0)), np.nan)
  
      def test_constant(self):
          # Constant array always gives 0
          x = np.ones((7, 4))
          assert_equal(stats.iqr(x), 0.0)
          assert_array_equal(stats.iqr(x, axis=0), np.zeros(4))
          assert_array_equal(stats.iqr(x, axis=1), np.zeros(7))
          assert_equal(stats.iqr(x, interpolation='linear'), 0.0)
          assert_equal(stats.iqr(x, interpolation='midpoint'), 0.0)
          assert_equal(stats.iqr(x, interpolation='nearest'), 0.0)
          assert_equal(stats.iqr(x, interpolation='lower'), 0.0)
          assert_equal(stats.iqr(x, interpolation='higher'), 0.0)
  
          # 0 only along constant dimensions
          # This also tests much of `axis`
          y = np.ones((4, 5, 6)) * np.arange(6)
          assert_array_equal(stats.iqr(y, axis=0), np.zeros((5, 6)))
          assert_array_equal(stats.iqr(y, axis=1), np.zeros((4, 6)))
          assert_array_equal(stats.iqr(y, axis=2), np.full((4, 5), 2.5))
          assert_array_equal(stats.iqr(y, axis=(0, 1)), np.zeros(6))
          assert_array_equal(stats.iqr(y, axis=(0, 2)), np.full(5, 3.))
          assert_array_equal(stats.iqr(y, axis=(1, 2)), np.full(4, 3.))
  
      def test_scalarlike(self):
          x = np.arange(1) + 7.0
          assert_equal(stats.iqr(x[0]), 0.0)
          assert_equal(stats.iqr(x), 0.0)
          assert_array_equal(stats.iqr(x, keepdims=True), [0.0])
  
      def test_2D(self):
          x = np.arange(15).reshape((3, 5))
          assert_equal(stats.iqr(x), 7.0)
          assert_array_equal(stats.iqr(x, axis=0), np.full(5, 5.))
          assert_array_equal(stats.iqr(x, axis=1), np.full(3, 2.))
          assert_array_equal(stats.iqr(x, axis=(0, 1)), 7.0)
          assert_array_equal(stats.iqr(x, axis=(1, 0)), 7.0)
  
      def test_axis(self):
          # The `axis` keyword is also put through its paces in `test_keepdims`.
          o = np.random.normal(size=(71, 23))
          x = np.dstack([o] * 10)                 # x.shape = (71, 23, 10)
          q = stats.iqr(o)
  
          assert_equal(stats.iqr(x, axis=(0, 1)), q)
          x = np.moveaxis(x, -1, 0)               # x.shape = (10, 71, 23)
          assert_equal(stats.iqr(x, axis=(2, 1)), q)
          x = x.swapaxes(0, 1)                    # x.shape = (71, 10, 23)
          assert_equal(stats.iqr(x, axis=(0, 2)), q)
          x = x.swapaxes(0, 1)                    # x.shape = (10, 71, 23)
  
          assert_equal(stats.iqr(x, axis=(0, 1, 2)),
                       stats.iqr(x, axis=None))
          assert_equal(stats.iqr(x, axis=(0,)),
                       stats.iqr(x, axis=0))
  
          d = np.arange(3 * 5 * 7 * 11)
          # Older versions of numpy only shuffle along axis=0.
          # Not sure about newer, don't care.
          np.random.shuffle(d)
          d = d.reshape((3, 5, 7, 11))
          assert_equal(stats.iqr(d, axis=(0, 1, 2))[0],
                       stats.iqr(d[:,:,:, 0].ravel()))
          assert_equal(stats.iqr(d, axis=(0, 1, 3))[1],
                       stats.iqr(d[:,:, 1,:].ravel()))
          assert_equal(stats.iqr(d, axis=(3, 1, -4))[2],
                       stats.iqr(d[:,:, 2,:].ravel()))
          assert_equal(stats.iqr(d, axis=(3, 1, 2))[2],
                       stats.iqr(d[2,:,:,:].ravel()))
          assert_equal(stats.iqr(d, axis=(3, 2))[2, 1],
                       stats.iqr(d[2, 1,:,:].ravel()))
          assert_equal(stats.iqr(d, axis=(1, -2))[2, 1],
                       stats.iqr(d[2, :, :, 1].ravel()))
          assert_equal(stats.iqr(d, axis=(1, 3))[2, 2],
                       stats.iqr(d[2, :, 2,:].ravel()))
  
          assert_raises(np.AxisError, stats.iqr, d, axis=4)
          assert_raises(ValueError, stats.iqr, d, axis=(0, 0))
  
      def test_rng(self):
          x = np.arange(5)
          assert_equal(stats.iqr(x), 2)
          assert_equal(stats.iqr(x, rng=(25, 87.5)), 2.5)
          assert_equal(stats.iqr(x, rng=(12.5, 75)), 2.5)
          assert_almost_equal(stats.iqr(x, rng=(10, 50)), 1.6)  # 3-1.4
  
          assert_raises(ValueError, stats.iqr, x, rng=(0, 101))
          assert_raises(ValueError, stats.iqr, x, rng=(np.nan, 25))
          assert_raises(TypeError, stats.iqr, x, rng=(0, 50, 60))
  
      def test_interpolation(self):
          x = np.arange(5)
          y = np.arange(4)
          # Default
          assert_equal(stats.iqr(x), 2)
          assert_equal(stats.iqr(y), 1.5)
          # Linear
          assert_equal(stats.iqr(x, interpolation='linear'), 2)
          assert_equal(stats.iqr(y, interpolation='linear'), 1.5)
          # Higher
          assert_equal(stats.iqr(x, interpolation='higher'), 2)
          assert_equal(stats.iqr(x, rng=(25, 80), interpolation='higher'), 3)
          assert_equal(stats.iqr(y, interpolation='higher'), 2)
          # Lower (will generally, but not always be the same as higher)
          assert_equal(stats.iqr(x, interpolation='lower'), 2)
          assert_equal(stats.iqr(x, rng=(25, 80), interpolation='lower'), 2)
          assert_equal(stats.iqr(y, interpolation='lower'), 2)
          # Nearest
          assert_equal(stats.iqr(x, interpolation='nearest'), 2)
          assert_equal(stats.iqr(y, interpolation='nearest'), 1)
          # Midpoint
          assert_equal(stats.iqr(x, interpolation='midpoint'), 2)
          assert_equal(stats.iqr(x, rng=(25, 80), interpolation='midpoint'), 2.5)
          assert_equal(stats.iqr(y, interpolation='midpoint'), 2)
  
          # Check all method= values new in numpy 1.22.0 are accepted
          if NumpyVersion(np.__version__) >= '1.22.0':
              for method in ('inverted_cdf', 'averaged_inverted_cdf',
                             'closest_observation', 'interpolated_inverted_cdf',
                             'hazen', 'weibull', 'median_unbiased',
                             'normal_unbiased'):
                  stats.iqr(y, interpolation=method)
  
          assert_raises(ValueError, stats.iqr, x, interpolation='foobar')
  
      def test_keepdims(self):
          # Also tests most of `axis`
          x = np.ones((3, 5, 7, 11))
          assert_equal(stats.iqr(x, axis=None, keepdims=False).shape, ())
          assert_equal(stats.iqr(x, axis=2, keepdims=False).shape, (3, 5, 11))
          assert_equal(stats.iqr(x, axis=(0, 1), keepdims=False).shape, (7, 11))
          assert_equal(stats.iqr(x, axis=(0, 3), keepdims=False).shape, (5, 7))
          assert_equal(stats.iqr(x, axis=(1,), keepdims=False).shape, (3, 7, 11))
          assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=False).shape, ())
          assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=False).shape, (7,))
  
          assert_equal(stats.iqr(x, axis=None, keepdims=True).shape, (1, 1, 1, 1))
          assert_equal(stats.iqr(x, axis=2, keepdims=True).shape, (3, 5, 1, 11))
          assert_equal(stats.iqr(x, axis=(0, 1), keepdims=True).shape, (1, 1, 7, 11))
          assert_equal(stats.iqr(x, axis=(0, 3), keepdims=True).shape, (1, 5, 7, 1))
          assert_equal(stats.iqr(x, axis=(1,), keepdims=True).shape, (3, 1, 7, 11))
          assert_equal(stats.iqr(x, (0, 1, 2, 3), keepdims=True).shape, (1, 1, 1, 1))
          assert_equal(stats.iqr(x, axis=(0, 1, 3), keepdims=True).shape, (1, 1, 7, 1))
  
      def test_nanpolicy(self):
          x = np.arange(15.0).reshape((3, 5))
  
          # No NaNs
          assert_equal(stats.iqr(x, nan_policy='propagate'), 7)
          assert_equal(stats.iqr(x, nan_policy='omit'), 7)
          assert_equal(stats.iqr(x, nan_policy='raise'), 7)
  
          # Yes NaNs
          x[1, 2] = np.nan
          with warnings.catch_warnings(record=True):
              warnings.simplefilter("always")
              assert_equal(stats.iqr(x, nan_policy='propagate'), np.nan)
              assert_equal(stats.iqr(x, axis=0, nan_policy='propagate'), [5, 5, np.nan, 5, 5])
              assert_equal(stats.iqr(x, axis=1, nan_policy='propagate'), [2, np.nan, 2])
  
          with warnings.catch_warnings(record=True):
              warnings.simplefilter("always")
              assert_equal(stats.iqr(x, nan_policy='omit'), 7.5)
              assert_equal(stats.iqr(x, axis=0, nan_policy='omit'), np.full(5, 5))
              assert_equal(stats.iqr(x, axis=1, nan_policy='omit'), [2, 2.5, 2])
  
          assert_raises(ValueError, stats.iqr, x, nan_policy='raise')
          assert_raises(ValueError, stats.iqr, x, axis=0, nan_policy='raise')
          assert_raises(ValueError, stats.iqr, x, axis=1, nan_policy='raise')
  
          # Bad policy
          assert_raises(ValueError, stats.iqr, x, nan_policy='barfood')
  
      def test_scale(self):
          x = np.arange(15.0).reshape((3, 5))
  
          # No NaNs
          assert_equal(stats.iqr(x, scale=1.0), 7)
          assert_almost_equal(stats.iqr(x, scale='normal'), 7 / 1.3489795)
          assert_equal(stats.iqr(x, scale=2.0), 3.5)
  
          # Yes NaNs
          x[1, 2] = np.nan
          with warnings.catch_warnings(record=True):
              warnings.simplefilter("always")
              assert_equal(stats.iqr(x, scale=1.0, nan_policy='propagate'), np.nan)
              assert_equal(stats.iqr(x, scale='normal', nan_policy='propagate'), np.nan)
              assert_equal(stats.iqr(x, scale=2.0, nan_policy='propagate'), np.nan)
              # axis=1 chosen to show behavior with both nans and without
              assert_equal(stats.iqr(x, axis=1, scale=1.0,
                                     nan_policy='propagate'), [2, np.nan, 2])
              assert_almost_equal(stats.iqr(x, axis=1, scale='normal',
                                            nan_policy='propagate'),
                                  np.array([2, np.nan, 2]) / 1.3489795)
              assert_equal(stats.iqr(x, axis=1, scale=2.0, nan_policy='propagate'),
                           [1, np.nan, 1])
              # Since NumPy 1.17.0.dev, warnings are no longer emitted by
              # np.percentile with nans, so we don't check the number of
              # warnings here. See https://github.com/numpy/numpy/pull/12679.
  
          assert_equal(stats.iqr(x, scale=1.0, nan_policy='omit'), 7.5)
          assert_almost_equal(stats.iqr(x, scale='normal', nan_policy='omit'),
                              7.5 / 1.3489795)
          assert_equal(stats.iqr(x, scale=2.0, nan_policy='omit'), 3.75)
  
          # Bad scale
          assert_raises(ValueError, stats.iqr, x, scale='foobar')
  
  
  class TestMoments:
      """
          Comparison numbers are found using R v.1.5.1
          note that length(testcase) = 4
          testmathworks comes from documentation for the
          Statistics Toolbox for Matlab and can be found at both
          https://www.mathworks.com/help/stats/kurtosis.html
          https://www.mathworks.com/help/stats/skewness.html
          Note that both test cases came from here.
      """
      testcase = [1,2,3,4]
      scalar_testcase = 4.
      np.random.seed(1234)
      testcase_moment_accuracy = np.random.rand(42)
      testmathworks = [1.165, 0.6268, 0.0751, 0.3516, -0.6965]
  
      def _assert_equal(self, actual, expect, *, shape=None, dtype=None):
          expect = np.asarray(expect)
          if shape is not None:
              expect = np.broadcast_to(expect, shape)
          assert_array_equal(actual, expect)
          if dtype is None:
              dtype = expect.dtype
          assert actual.dtype == dtype
  
      def test_moment(self):
          # mean((testcase-mean(testcase))**power,axis=0),axis=0))**power))
          y = stats.moment(self.scalar_testcase)
          assert_approx_equal(y, 0.0)
          y = stats.moment(self.testcase, 0)
          assert_approx_equal(y, 1.0)
          y = stats.moment(self.testcase, 1)
          assert_approx_equal(y, 0.0, 10)
          y = stats.moment(self.testcase, 2)
          assert_approx_equal(y, 1.25)
          y = stats.moment(self.testcase, 3)
          assert_approx_equal(y, 0.0)
          y = stats.moment(self.testcase, 4)
          assert_approx_equal(y, 2.5625)
  
          # check array_like input for moment
          y = stats.moment(self.testcase, [1, 2, 3, 4])
          assert_allclose(y, [0, 1.25, 0, 2.5625])
  
          # check moment input consists only of integers
          y = stats.moment(self.testcase, 0.0)
          assert_approx_equal(y, 1.0)
          assert_raises(ValueError, stats.moment, self.testcase, 1.2)
          y = stats.moment(self.testcase, [1.0, 2, 3, 4.0])
          assert_allclose(y, [0, 1.25, 0, 2.5625])
  
          # test empty input
          y = stats.moment([])
          self._assert_equal(y, np.nan, dtype=np.float64)
          y = stats.moment(np.array([], dtype=np.float32))
          self._assert_equal(y, np.nan, dtype=np.float32)
          y = stats.moment(np.zeros((1, 0)), axis=0)
          self._assert_equal(y, [], shape=(0,), dtype=np.float64)
          y = stats.moment([[]], axis=1)
          self._assert_equal(y, np.nan, shape=(1,), dtype=np.float64)
          y = stats.moment([[]], moment=[0, 1], axis=0)
          self._assert_equal(y, [], shape=(2, 0))
  
          x = np.arange(10.)
          x[9] = np.nan
          assert_equal(stats.moment(x, 2), np.nan)
          assert_almost_equal(stats.moment(x, nan_policy='omit'), 0.0)
          assert_raises(ValueError, stats.moment, x, nan_policy='raise')
          assert_raises(ValueError, stats.moment, x, nan_policy='foobar')
  
      @pytest.mark.parametrize('dtype', [np.float32, np.float64, np.complex128])
      @pytest.mark.parametrize('expect, moment', [(0, 1), (1, 0)])
      def test_constant_moments(self, dtype, expect, moment):
          x = np.random.rand(5).astype(dtype)
          y = stats.moment(x, moment=moment)
          self._assert_equal(y, expect, dtype=dtype)
  
          y = stats.moment(np.broadcast_to(x, (6, 5)), axis=0, moment=moment)
          self._assert_equal(y, expect, shape=(5,), dtype=dtype)
  
          y = stats.moment(np.broadcast_to(x, (1, 2, 3, 4, 5)), axis=2,
                           moment=moment)
          self._assert_equal(y, expect, shape=(1, 2, 4, 5), dtype=dtype)
  
          y = stats.moment(np.broadcast_to(x, (1, 2, 3, 4, 5)), axis=None,
                           moment=moment)
          self._assert_equal(y, expect, shape=(), dtype=dtype)
  
      def test_moment_propagate_nan(self):
          # Check that the shape of the result is the same for inputs
          # with and without nans, cf gh-5817
          a = np.arange(8).reshape(2, -1).astype(float)
          a[1, 0] = np.nan
          mm = stats.moment(a, 2, axis=1, nan_policy="propagate")
          np.testing.assert_allclose(mm, [1.25, np.nan], atol=1e-15)
  
      def test_moment_empty_moment(self):
          # tests moment with empty `moment` list
          with pytest.raises(ValueError, match=r"'moment' must be a scalar or a"
                                               r" non-empty 1D list/array."):
              stats.moment([1, 2, 3, 4], moment=[])
  
      def test_skewness(self):
          # Scalar test case
          with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              y = stats.skew(self.scalar_testcase)
              assert np.isnan(y)
          # sum((testmathworks-mean(testmathworks,axis=0))**3,axis=0) /
          #     ((sqrt(var(testmathworks)*4/5))**3)/5
          y = stats.skew(self.testmathworks)
          assert_approx_equal(y, -0.29322304336607, 10)
          y = stats.skew(self.testmathworks, bias=0)
          assert_approx_equal(y, -0.437111105023940, 10)
          y = stats.skew(self.testcase)
          assert_approx_equal(y, 0.0, 10)
  
          x = np.arange(10.)
          x[9] = np.nan
          with np.errstate(invalid='ignore'):
              assert_equal(stats.skew(x), np.nan)
          assert_equal(stats.skew(x, nan_policy='omit'), 0.)
          assert_raises(ValueError, stats.skew, x, nan_policy='raise')
          assert_raises(ValueError, stats.skew, x, nan_policy='foobar')
  
      def test_skewness_scalar(self):
          # `skew` must return a scalar for 1-dim input
          assert_equal(stats.skew(arange(10)), 0.0)
  
      def test_skew_propagate_nan(self):
          # Check that the shape of the result is the same for inputs
          # with and without nans, cf gh-5817
          a = np.arange(8).reshape(2, -1).astype(float)
          a[1, 0] = np.nan
          with np.errstate(invalid='ignore'):
              s = stats.skew(a, axis=1, nan_policy="propagate")
          np.testing.assert_allclose(s, [0, np.nan], atol=1e-15)
  
      def test_skew_constant_value(self):
          # Skewness of a constant input should be zero even when the mean is not
          # exact (gh-13245)
          with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              a = np.repeat(-0.27829495, 10)
              assert np.isnan(stats.skew(a))
              assert np.isnan(stats.skew(a * float(2**50)))
              assert np.isnan(stats.skew(a / float(2**50)))
              assert np.isnan(stats.skew(a, bias=False))
  
              # similarly, from gh-11086:
              assert np.isnan(stats.skew([14.3]*7))
              assert np.isnan(stats.skew(1 + np.arange(-3, 4)*1e-16))
  
      def test_kurtosis(self):
          # Scalar test case
          with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              y = stats.kurtosis(self.scalar_testcase)
              assert np.isnan(y)
          #   sum((testcase-mean(testcase,axis=0))**4,axis=0)/((sqrt(var(testcase)*3/4))**4)/4
          #   sum((test2-mean(testmathworks,axis=0))**4,axis=0)/((sqrt(var(testmathworks)*4/5))**4)/5
          #   Set flags for axis = 0 and
          #   fisher=0 (Pearson's defn of kurtosis for compatibility with Matlab)
          y = stats.kurtosis(self.testmathworks, 0, fisher=0, bias=1)
          assert_approx_equal(y, 2.1658856802973, 10)
  
          # Note that MATLAB has confusing docs for the following case
          #  kurtosis(x,0) gives an unbiased estimate of Pearson's skewness
          #  kurtosis(x)  gives a biased estimate of Fisher's skewness (Pearson-3)
          #  The MATLAB docs imply that both should give Fisher's
          y = stats.kurtosis(self.testmathworks, fisher=0, bias=0)
          assert_approx_equal(y, 3.663542721189047, 10)
          y = stats.kurtosis(self.testcase, 0, 0)
          assert_approx_equal(y, 1.64)
  
          x = np.arange(10.)
          x[9] = np.nan
          assert_equal(stats.kurtosis(x), np.nan)
          assert_almost_equal(stats.kurtosis(x, nan_policy='omit'), -1.230000)
          assert_raises(ValueError, stats.kurtosis, x, nan_policy='raise')
          assert_raises(ValueError, stats.kurtosis, x, nan_policy='foobar')
  
      def test_kurtosis_array_scalar(self):
          assert_equal(type(stats.kurtosis([1,2,3])), float)
  
      def test_kurtosis_propagate_nan(self):
          # Check that the shape of the result is the same for inputs
          # with and without nans, cf gh-5817
          a = np.arange(8).reshape(2, -1).astype(float)
          a[1, 0] = np.nan
          k = stats.kurtosis(a, axis=1, nan_policy="propagate")
          np.testing.assert_allclose(k, [-1.36, np.nan], atol=1e-15)
  
      def test_kurtosis_constant_value(self):
          # Kurtosis of a constant input should be zero, even when the mean is not
          # exact (gh-13245)
          a = np.repeat(-0.27829495, 10)
          with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              assert np.isnan(stats.kurtosis(a, fisher=False))
              assert np.isnan(stats.kurtosis(a * float(2**50), fisher=False))
              assert np.isnan(stats.kurtosis(a / float(2**50), fisher=False))
              assert np.isnan(stats.kurtosis(a, fisher=False, bias=False))
  
      def test_moment_accuracy(self):
          # 'moment' must have a small enough error compared to the slower
          #  but very accurate numpy.power() implementation.
          tc_no_mean = self.testcase_moment_accuracy - \
                       np.mean(self.testcase_moment_accuracy)
          assert_allclose(np.power(tc_no_mean, 42).mean(),
                              stats.moment(self.testcase_moment_accuracy, 42))
  
      def test_precision_loss_gh15554(self):
          # gh-15554 was one of several issues that have reported problems with
          # constant or near-constant input. We can't always fix these, but
          # make sure there's a warning.
          with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              rng = np.random.default_rng(34095309370)
              a = rng.random(size=(100, 10))
              a[:, 0] = 1.01
              stats.skew(a)[0]
  
  
  class TestStudentTest:
      X1 = np.array([-1, 0, 1])
      X2 = np.array([0, 1, 2])
      T1_0 = 0
      P1_0 = 1
      T1_1 = -1.7320508075
      P1_1 = 0.22540333075
      T1_2 = -3.464102
      P1_2 = 0.0741799
      T2_0 = 1.732051
      P2_0 = 0.2254033
      P1_1_l = P1_1 / 2
      P1_1_g = 1 - (P1_1 / 2)
  
      def test_onesample(self):
          with suppress_warnings() as sup, np.errstate(invalid="ignore"), \
                  pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
              t, p = stats.ttest_1samp(4., 3.)
          assert_(np.isnan(t))
          assert_(np.isnan(p))
  
          t, p = stats.ttest_1samp(self.X1, 0)
  
          assert_array_almost_equal(t, self.T1_0)
          assert_array_almost_equal(p, self.P1_0)
  
          res = stats.ttest_1samp(self.X1, 0)
          attributes = ('statistic', 'pvalue')
          check_named_results(res, attributes)
  
          t, p = stats.ttest_1samp(self.X2, 0)
  
          assert_array_almost_equal(t, self.T2_0)
          assert_array_almost_equal(p, self.P2_0)
  
          t, p = stats.ttest_1samp(self.X1, 1)
  
          assert_array_almost_equal(t, self.T1_1)
          assert_array_almost_equal(p, self.P1_1)
  
          t, p = stats.ttest_1samp(self.X1, 2)
  
          assert_array_almost_equal(t, self.T1_2)
          assert_array_almost_equal(p, self.P1_2)
  
          # check nan policy
          x = stats.norm.rvs(loc=5, scale=10, size=51, random_state=7654567)
          x[50] = np.nan
          with np.errstate(invalid="ignore"):
              assert_array_equal(stats.ttest_1samp(x, 5.0), (np.nan, np.nan))
  
              assert_array_almost_equal(stats.ttest_1samp(x, 5.0, nan_policy='omit'),
                                        (-1.6412624074367159, 0.107147027334048005))
              assert_raises(ValueError, stats.ttest_1samp, x, 5.0, nan_policy='raise')
              assert_raises(ValueError, stats.ttest_1samp, x, 5.0,
                            nan_policy='foobar')
  
      def test_1samp_alternative(self):
          assert_raises(ValueError, stats.ttest_1samp, self.X1, 0,
                        alternative="error")
  
          t, p = stats.ttest_1samp(self.X1, 1, alternative="less")
          assert_allclose(p, self.P1_1_l)
          assert_allclose(t, self.T1_1)
  
          t, p = stats.ttest_1samp(self.X1, 1, alternative="greater")
          assert_allclose(p, self.P1_1_g)
          assert_allclose(t, self.T1_1)
  
  
  class TestPercentileOfScore:
  
      def f(self, *args, **kwargs):
          return stats.percentileofscore(*args, **kwargs)
  
      @pytest.mark.parametrize("kind, result", [("rank", 40),
                                                ("mean", 35),
                                                ("strict", 30),
                                                ("weak", 40)])
      def test_unique(self, kind, result):
          a = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
          assert_equal(self.f(a, 4, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", 45),
                                                ("mean", 40),
                                                ("strict", 30),
                                                ("weak", 50)])
      def test_multiple2(self, kind, result):
          a = [1, 2, 3, 4, 4, 5, 6, 7, 8, 9]
          assert_equal(self.f(a, 4, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", 50),
                                                ("mean", 45),
                                                ("strict", 30),
                                                ("weak", 60)])
      def test_multiple3(self, kind, result):
          a = [1, 2, 3, 4, 4, 4, 5, 6, 7, 8]
          assert_equal(self.f(a, 4, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", 30),
                                                ("mean", 30),
                                                ("strict", 30),
                                                ("weak", 30)])
      def test_missing(self, kind, result):
          a = [1, 2, 3, 5, 6, 7, 8, 9, 10, 11]
          assert_equal(self.f(a, 4, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", 40),
                                                ("mean", 35),
                                                ("strict", 30),
                                                ("weak", 40)])
      def test_large_numbers(self, kind, result):
          a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
          assert_equal(self.f(a, 40, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", 50),
                                                ("mean", 45),
                                                ("strict", 30),
                                                ("weak", 60)])
      def test_large_numbers_multiple3(self, kind, result):
          a = [10, 20, 30, 40, 40, 40, 50, 60, 70, 80]
          assert_equal(self.f(a, 40, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", 30),
                                                ("mean", 30),
                                                ("strict", 30),
                                                ("weak", 30)])
      def test_large_numbers_missing(self, kind, result):
          a = [10, 20, 30, 50, 60, 70, 80, 90, 100, 110]
          assert_equal(self.f(a, 40, kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", [0, 10, 100, 100]),
                                                ("mean", [0, 5, 95, 100]),
                                                ("strict", [0, 0, 90, 100]),
                                                ("weak", [0, 10, 100, 100])])
      def test_boundaries(self, kind, result):
          a = [10, 20, 30, 50, 60, 70, 80, 90, 100, 110]
          assert_equal(self.f(a, [0, 10, 110, 200], kind=kind), result)
  
      @pytest.mark.parametrize("kind, result", [("rank", [0, 10, 100]),
                                                ("mean", [0, 5, 95]),
                                                ("strict", [0, 0, 90]),
                                                ("weak", [0, 10, 100])])
      def test_inf(self, kind, result):
          a = [1, 2, 3, 4, 5, 6, 7, 8, 9, +np.inf]
          assert_equal(self.f(a, [-np.inf, 1, +np.inf], kind=kind), result)
  
      cases = [("propagate", [], 1, np.nan),
               ("propagate", [np.nan], 1, np.nan),
               ("propagate", [np.nan], [0, 1, 2], [np.nan, np.nan, np.nan]),
               ("propagate", [1, 2], [1, 2, np.nan], [50, 100, np.nan]),
               ("omit", [1, 2, np.nan], [0, 1, 2], [0, 50, 100]),
               ("omit", [1, 2], [0, 1, np.nan], [0, 50, np.nan]),
               ("omit", [np.nan, np.nan], [0, 1, 2], [np.nan, np.nan, np.nan])]
  
      @pytest.mark.parametrize("policy, a, score, result", cases)
      def test_nans_ok(self, policy, a, score, result):
          assert_equal(self.f(a, score, nan_policy=policy), result)
  
      cases = [
          ("raise", [1, 2, 3, np.nan], [1, 2, 3],
           "The input contains nan values"),
          ("raise", [1, 2, 3], [1, 2, 3, np.nan],
           "The input contains nan values"),
      ]
  
      @pytest.mark.parametrize("policy, a, score, message", cases)
      def test_nans_fail(self, policy, a, score, message):
          with assert_raises(ValueError, match=message):
              self.f(a, score, nan_policy=policy)
  
      @pytest.mark.parametrize("shape", [
          (6, ),
          (2, 3),
          (2, 1, 3),
          (2, 1, 1, 3),
      ])
      def test_nd(self, shape):
          a = np.array([0, 1, 2, 3, 4, 5])
          scores = a.reshape(shape)
          results = scores*10
          a = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
          assert_equal(self.f(a, scores, kind="rank"), results)
  
  
  PowerDivCase = namedtuple('Case',  # type: ignore[name-match]
                            ['f_obs', 'f_exp', 'ddof', 'axis',
                             'chi2',     # Pearson's
                             'log',      # G-test (log-likelihood)
                             'mod_log',  # Modified log-likelihood
                             'cr',       # Cressie-Read (lambda=2/3)
                             ])
  
  # The details of the first two elements in power_div_1d_cases are used
  # in a test in TestPowerDivergence.  Check that code before making
  # any changes here.
  power_div_1d_cases = [
      # Use the default f_exp.
      PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=None, ddof=0, axis=None,
                   chi2=4,
                   log=2*(4*np.log(4/8) + 12*np.log(12/8)),
                   mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
                   cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
      # Give a non-uniform f_exp.
      PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=[2, 16, 12, 2], ddof=0, axis=None,
                   chi2=24,
                   log=2*(4*np.log(4/2) + 8*np.log(8/16) + 8*np.log(8/2)),
                   mod_log=2*(2*np.log(2/4) + 16*np.log(16/8) + 2*np.log(2/8)),
                   cr=(4*((4/2)**(2/3) - 1) + 8*((8/16)**(2/3) - 1) +
                       8*((8/2)**(2/3) - 1))/(5/9)),
      # f_exp is a scalar.
      PowerDivCase(f_obs=[4, 8, 12, 8], f_exp=8, ddof=0, axis=None,
                   chi2=4,
                   log=2*(4*np.log(4/8) + 12*np.log(12/8)),
                   mod_log=2*(8*np.log(8/4) + 8*np.log(8/12)),
                   cr=(4*((4/8)**(2/3) - 1) + 12*((12/8)**(2/3) - 1))/(5/9)),
      # f_exp equal to f_obs.
      PowerDivCase(f_obs=[3, 5, 7, 9], f_exp=[3, 5, 7, 9], ddof=0, axis=0,
                   chi2=0, log=0, mod_log=0, cr=0),
  ]
  
  
  power_div_empty_cases = [
      # Shape is (0,)--a data set with length 0.  The computed
      # test statistic should be 0.
      PowerDivCase(f_obs=[],
                   f_exp=None, ddof=0, axis=0,
                   chi2=0, log=0, mod_log=0, cr=0),
      # Shape is (0, 3).  This is 3 data sets, but each data set has
      # length 0, so the computed test statistic should be [0, 0, 0].
      PowerDivCase(f_obs=np.array([[],[],[]]).T,
                   f_exp=None, ddof=0, axis=0,
                   chi2=[0, 0, 0],
                   log=[0, 0, 0],
                   mod_log=[0, 0, 0],
                   cr=[0, 0, 0]),
      # Shape is (3, 0).  This represents an empty collection of
      # data sets in which each data set has length 3.  The test
      # statistic should be an empty array.
      PowerDivCase(f_obs=np.array([[],[],[]]),
                   f_exp=None, ddof=0, axis=0,
                   chi2=[],
                   log=[],
                   mod_log=[],
                   cr=[]),
  ]
  
  
  class TestPowerDivergence:
  
      def check_power_divergence(self, f_obs, f_exp, ddof, axis, lambda_,
                                 expected_stat):
          f_obs = np.asarray(f_obs)
          if axis is None:
              num_obs = f_obs.size
          else:
              b = np.broadcast(f_obs, f_exp)
              num_obs = b.shape[axis]
  
          with suppress_warnings() as sup:
              sup.filter(RuntimeWarning, "Mean of empty slice")
              stat, p = stats.power_divergence(
                                  f_obs=f_obs, f_exp=f_exp, ddof=ddof,
                                  axis=axis, lambda_=lambda_)
              assert_allclose(stat, expected_stat)
  
              if lambda_ == 1 or lambda_ == "pearson":
                  # Also test stats.chisquare.
                  stat, p = stats.chisquare(f_obs=f_obs, f_exp=f_exp, ddof=ddof,
                                            axis=axis)
                  assert_allclose(stat, expected_stat)
  
          ddof = np.asarray(ddof)
          expected_p = stats.distributions.chi2.sf(expected_stat,
                                                   num_obs - 1 - ddof)
          assert_allclose(p, expected_p)
  
      def test_basic(self):
          for case in power_div_1d_cases:
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     None, case.chi2)
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     "pearson", case.chi2)
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     1, case.chi2)
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     "log-likelihood", case.log)
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     "mod-log-likelihood", case.mod_log)
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     "cressie-read", case.cr)
              self.check_power_divergence(
                     case.f_obs, case.f_exp, case.ddof, case.axis,
                     2/3, case.cr)
  
      def test_basic_masked(self):
          for case in power_div_1d_cases:
              mobs = np.ma.array(case.f_obs)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     None, case.chi2)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     "pearson", case.chi2)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     1, case.chi2)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     "log-likelihood", case.log)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     "mod-log-likelihood", case.mod_log)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     "cressie-read", case.cr)
              self.check_power_divergence(
                     mobs, case.f_exp, case.ddof, case.axis,
                     2/3, case.cr)
  
      def test_axis(self):
          case0 = power_div_1d_cases[0]
          case1 = power_div_1d_cases[1]
          f_obs = np.vstack((case0.f_obs, case1.f_obs))
          f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
                             case1.f_exp))
          # Check the four computational code paths in power_divergence
          # using a 2D array with axis=1.
          self.check_power_divergence(
                 f_obs, f_exp, 0, 1,
                 "pearson", [case0.chi2, case1.chi2])
          self.check_power_divergence(
                 f_obs, f_exp, 0, 1,
                 "log-likelihood", [case0.log, case1.log])
          self.check_power_divergence(
                 f_obs, f_exp, 0, 1,
                 "mod-log-likelihood", [case0.mod_log, case1.mod_log])
          self.check_power_divergence(
                 f_obs, f_exp, 0, 1,
                 "cressie-read", [case0.cr, case1.cr])
          # Reshape case0.f_obs to shape (2,2), and use axis=None.
          # The result should be the same.
          self.check_power_divergence(
                 np.array(case0.f_obs).reshape(2, 2), None, 0, None,
                 "pearson", case0.chi2)
  
      def test_ddof_broadcasting(self):
          # Test that ddof broadcasts correctly.
          # ddof does not affect the test statistic.  It is broadcast
          # with the computed test statistic for the computation of
          # the p value.
  
          case0 = power_div_1d_cases[0]
          case1 = power_div_1d_cases[1]
          # Create 4x2 arrays of observed and expected frequencies.
          f_obs = np.vstack((case0.f_obs, case1.f_obs)).T
          f_exp = np.vstack((np.ones_like(case0.f_obs)*np.mean(case0.f_obs),
                             case1.f_exp)).T
  
          expected_chi2 = [case0.chi2, case1.chi2]
  
          # ddof has shape (2, 1).  This is broadcast with the computed
          # statistic, so p will have shape (2,2).
          ddof = np.array([[0], [1]])
  
          stat, p = stats.power_divergence(f_obs, f_exp, ddof=ddof)
          assert_allclose(stat, expected_chi2)
  
          # Compute the p values separately, passing in scalars for ddof.
          stat0, p0 = stats.power_divergence(f_obs, f_exp, ddof=ddof[0,0])
          stat1, p1 = stats.power_divergence(f_obs, f_exp, ddof=ddof[1,0])
  
          assert_array_equal(p, np.vstack((p0, p1)))
  
      def test_empty_cases(self):
          with warnings.catch_warnings():
              for case in power_div_empty_cases:
                  self.check_power_divergence(
                         case.f_obs, case.f_exp, case.ddof, case.axis,
                         "pearson", case.chi2)
                  self.check_power_divergence(
                         case.f_obs, case.f_exp, case.ddof, case.axis,
                         "log-likelihood", case.log)
                  self.check_power_divergence(
                         case.f_obs, case.f_exp, case.ddof, case.axis,
                         "mod-log-likelihood", case.mod_log)
                  self.check_power_divergence(
                         case.f_obs, case.f_exp, case.ddof, case.axis,
                         "cressie-read", case.cr)
  
      def test_power_divergence_result_attributes(self):
          f_obs = power_div_1d_cases[0].f_obs
          f_exp = power_div_1d_cases[0].f_exp
          ddof = power_div_1d_cases[0].ddof
          axis = power_div_1d_cases[0].axis
  
          res = stats.power_divergence(f_obs=f_obs, f_exp=f_exp, ddof=ddof,
                                       axis=axis, lambda_="pearson")
          attributes = ('statistic', 'pvalue')
          check_named_results(res, attributes)
  
      def test_power_divergence_gh_12282(self):
          # The sums of observed and expected frequencies must match
          f_obs = np.array([[10, 20], [30, 20]])
          f_exp = np.array([[5, 15], [35, 25]])
          with assert_raises(ValueError, match='For each axis slice...'):
              stats.power_divergence(f_obs=[10, 20], f_exp=[30, 60])
          with assert_raises(ValueError, match='For each axis slice...'):
              stats.power_divergence(f_obs=f_obs, f_exp=f_exp, axis=1)
          stat, pval = stats.power_divergence(f_obs=f_obs, f_exp=f_exp)
          assert_allclose(stat, [5.71428571, 2.66666667])
          assert_allclose(pval, [0.01682741, 0.10247043])
  
  
  def test_gh_chisquare_12282():
      # Currently `chisquare` is implemented via power_divergence
      # in case that ever changes, perform a basic test like
      # test_power_divergence_gh_12282
      with assert_raises(ValueError, match='For each axis slice...'):
          stats.chisquare(f_obs=[10, 20], f_exp=[30, 60])
  
  
  @pytest.mark.parametrize("n, dtype", [(200, np.uint8), (1000000, np.int32)])
  def test_chiquare_data_types(n, dtype):
      # Regression test for gh-10159.
      obs = np.array([n, 0], dtype=dtype)
      exp = np.array([n // 2, n // 2], dtype=dtype)
      stat, p = stats.chisquare(obs, exp)
      assert_allclose(stat, n, rtol=1e-13)
  
  
  def test_chisquare_masked_arrays():
      # Test masked arrays.
      obs = np.array([[8, 8, 16, 32, -1], [-1, -1, 3, 4, 5]]).T
      mask = np.array([[0, 0, 0, 0, 1], [1, 1, 0, 0, 0]]).T
      mobs = np.ma.masked_array(obs, mask)
      expected_chisq = np.array([24.0, 0.5])
      expected_g = np.array([2*(2*8*np.log(0.5) + 32*np.log(2.0)),
                             2*(3*np.log(0.75) + 5*np.log(1.25))])
  
      chi2 = stats.distributions.chi2
  
      chisq, p = stats.chisquare(mobs)
      mat.assert_array_equal(chisq, expected_chisq)
      mat.assert_array_almost_equal(p, chi2.sf(expected_chisq,
                                               mobs.count(axis=0) - 1))
  
      g, p = stats.power_divergence(mobs, lambda_='log-likelihood')
      mat.assert_array_almost_equal(g, expected_g, decimal=15)
      mat.assert_array_almost_equal(p, chi2.sf(expected_g,
                                               mobs.count(axis=0) - 1))
  
      chisq, p = stats.chisquare(mobs.T, axis=1)
      mat.assert_array_equal(chisq, expected_chisq)
      mat.assert_array_almost_equal(p, chi2.sf(expected_chisq,
                                               mobs.T.count(axis=1) - 1))
      g, p = stats.power_divergence(mobs.T, axis=1, lambda_="log-likelihood")
      mat.assert_array_almost_equal(g, expected_g, decimal=15)
      mat.assert_array_almost_equal(p, chi2.sf(expected_g,
                                               mobs.count(axis=0) - 1))
  
      obs1 = np.ma.array([3, 5, 6, 99, 10], mask=[0, 0, 0, 1, 0])
      exp1 = np.ma.array([2, 4, 8, 10, 99], mask=[0, 0, 0, 0, 1])
      chi2, p = stats.chisquare(obs1, f_exp=exp1)
      # Because of the mask at index 3 of obs1 and at index 4 of exp1,
      # only the first three elements are included in the calculation
      # of the statistic.
      mat.assert_array_equal(chi2, 1/2 + 1/4 + 4/8)
  
      # When axis=None, the two values should have type np.float64.
      chisq, p = stats.chisquare(np.ma.array([1,2,3]), axis=None)
      assert_(isinstance(chisq, np.float64))
      assert_(isinstance(p, np.float64))
      assert_equal(chisq, 1.0)
      assert_almost_equal(p, stats.distributions.chi2.sf(1.0, 2))
  
      # Empty arrays:
      # A data set with length 0 returns a masked scalar.
      with np.errstate(invalid='ignore'):
          with suppress_warnings() as sup:
              sup.filter(RuntimeWarning, "Mean of empty slice")
              chisq, p = stats.chisquare(np.ma.array([]))
      assert_(isinstance(chisq, np.ma.MaskedArray))
      assert_equal(chisq.shape, ())
      assert_(chisq.mask)
  
      empty3 = np.ma.array([[],[],[]])
  
      # empty3 is a collection of 0 data sets (whose lengths would be 3, if
      # there were any), so the return value is an array with length 0.
      chisq, p = stats.chisquare(empty3)
      assert_(isinstance(chisq, np.ma.MaskedArray))
      mat.assert_array_equal(chisq, [])
  
      # empty3.T is an array containing 3 data sets, each with length 0,
      # so an array of size (3,) is returned, with all values masked.
      with np.errstate(invalid='ignore'):
          with suppress_warnings() as sup:
              sup.filter(RuntimeWarning, "Mean of empty slice")
              chisq, p = stats.chisquare(empty3.T)
  
      assert_(isinstance(chisq, np.ma.MaskedArray))
      assert_equal(chisq.shape, (3,))
      assert_(np.all(chisq.mask))
  
  
  def test_power_divergence_against_cressie_read_data():
      # Test stats.power_divergence against tables 4 and 5 from
      # Cressie and Read, "Multimonial Goodness-of-Fit Tests",
      # J. R. Statist. Soc. B (1984), Vol 46, No. 3, pp. 440-464.
      # This tests the calculation for several values of lambda.
  
      # Table 4 data recalculated for greater precision according to:
      # Shelby J. Haberman, Analysis of Qualitative Data: Volume 1
      # Introductory Topics, Academic Press, New York, USA (1978).
      obs = np.array([15, 11, 14, 17, 5, 11, 10, 4, 8,
                      10, 7, 9, 11, 3, 6, 1, 1, 4])
      beta = -0.083769  # Haberman (1978), p. 15
      i = np.arange(1, len(obs) + 1)
      alpha = np.log(obs.sum() / np.exp(beta*i).sum())
      expected_counts = np.exp(alpha + beta*i)
  
      # `table4` holds just the second and third columns from Table 4.
      table4 = np.vstack((obs, expected_counts)).T
  
      table5 = np.array([
          # lambda, statistic
          -10.0, 72.2e3,
          -5.0, 28.9e1,
          -3.0, 65.6,
          -2.0, 40.6,
          -1.5, 34.0,
          -1.0, 29.5,
          -0.5, 26.5,
          0.0, 24.6,
          0.5, 23.4,
          0.67, 23.1,
          1.0, 22.7,
          1.5, 22.6,
          2.0, 22.9,
          3.0, 24.8,
          5.0, 35.5,
          10.0, 21.4e1,
          ]).reshape(-1, 2)
  
      for lambda_, expected_stat in table5:
          stat, p = stats.power_divergence(table4[:,0], table4[:,1],
                                           lambda_=lambda_)
          assert_allclose(stat, expected_stat, rtol=5e-3)
  
  
  def test_friedmanchisquare():
      # see ticket:113
      # verified with matlab and R
      # From Demsar "Statistical Comparisons of Classifiers over Multiple Data Sets"
      # 2006, Xf=9.28 (no tie handling, tie corrected Xf >=9.28)
      x1 = [array([0.763, 0.599, 0.954, 0.628, 0.882, 0.936, 0.661, 0.583,
                   0.775, 1.0, 0.94, 0.619, 0.972, 0.957]),
            array([0.768, 0.591, 0.971, 0.661, 0.888, 0.931, 0.668, 0.583,
                   0.838, 1.0, 0.962, 0.666, 0.981, 0.978]),
            array([0.771, 0.590, 0.968, 0.654, 0.886, 0.916, 0.609, 0.563,
                   0.866, 1.0, 0.965, 0.614, 0.9751, 0.946]),
            array([0.798, 0.569, 0.967, 0.657, 0.898, 0.931, 0.685, 0.625,
                   0.875, 1.0, 0.962, 0.669, 0.975, 0.970])]
  
      # From "Bioestadistica para las ciencias de la salud" Xf=18.95 p<0.001:
      x2 = [array([4,3,5,3,5,3,2,5,4,4,4,3]),
            array([2,2,1,2,3,1,2,3,2,1,1,3]),
            array([2,4,3,3,4,3,3,4,4,1,2,1]),
            array([3,5,4,3,4,4,3,3,3,4,4,4])]
  
      # From Jerrorl H. Zar, "Biostatistical Analysis"(example 12.6), Xf=10.68, 0.005 < p < 0.01:
      # Probability from this example is inexact using Chisquare approximation of Friedman Chisquare.
      x3 = [array([7.0,9.9,8.5,5.1,10.3]),
            array([5.3,5.7,4.7,3.5,7.7]),
            array([4.9,7.6,5.5,2.8,8.4]),
            array([8.8,8.9,8.1,3.3,9.1])]
  
      assert_array_almost_equal(stats.friedmanchisquare(x1[0],x1[1],x1[2],x1[3]),
                                (10.2283464566929, 0.0167215803284414))
      assert_array_almost_equal(stats.friedmanchisquare(x2[0],x2[1],x2[2],x2[3]),
                                (18.9428571428571, 0.000280938375189499))
      assert_array_almost_equal(stats.friedmanchisquare(x3[0],x3[1],x3[2],x3[3]),
                                (10.68, 0.0135882729582176))
      assert_raises(ValueError, stats.friedmanchisquare,x3[0],x3[1])
  
      # test for namedtuple attribute results
      attributes = ('statistic', 'pvalue')
      res = stats.friedmanchisquare(*x1)
      check_named_results(res, attributes)
  
      # test using mstats
      assert_array_almost_equal(mstats.friedmanchisquare(x1[0], x1[1],
                                                         x1[2], x1[3]),
                                (10.2283464566929, 0.0167215803284414))
      # the following fails
      # assert_array_almost_equal(mstats.friedmanchisquare(x2[0],x2[1],x2[2],x2[3]),
      #                           (18.9428571428571, 0.000280938375189499))
      assert_array_almost_equal(mstats.friedmanchisquare(x3[0], x3[1],
                                                         x3[2], x3[3]),
                                (10.68, 0.0135882729582176))
      assert_raises(ValueError, mstats.friedmanchisquare,x3[0],x3[1])
  
  
  class TestKSTest:
      """Tests kstest and ks_1samp agree with K-S various sizes, alternatives, modes."""
  
      def _testOne(self, x, alternative, expected_statistic, expected_prob, mode='auto', decimal=14):
          result = stats.kstest(x, 'norm', alternative=alternative, mode=mode)
          expected = np.array([expected_statistic, expected_prob])
          assert_array_almost_equal(np.array(result), expected, decimal=decimal)
  
      def _test_kstest_and_ks1samp(self, x, alternative, mode='auto', decimal=14):
          result = stats.kstest(x, 'norm', alternative=alternative, mode=mode)
          result_1samp = stats.ks_1samp(x, stats.norm.cdf, alternative=alternative, mode=mode)
          assert_array_almost_equal(np.array(result), result_1samp, decimal=decimal)
  
      def test_namedtuple_attributes(self):
          x = np.linspace(-1, 1, 9)
          # test for namedtuple attribute results
          attributes = ('statistic', 'pvalue')
          res = stats.kstest(x, 'norm')
          check_named_results(res, attributes)
  
      def test_agree_with_ks_1samp(self):
          x = np.linspace(-1, 1, 9)
          self._test_kstest_and_ks1samp(x, 'two-sided')
  
          x = np.linspace(-15, 15, 9)
          self._test_kstest_and_ks1samp(x, 'two-sided')
  
          x = [-1.23, 0.06, -0.60, 0.17, 0.66, -0.17, -0.08, 0.27, -0.98, -0.99]
          self._test_kstest_and_ks1samp(x, 'two-sided')
          self._test_kstest_and_ks1samp(x, 'greater', mode='exact')
          self._test_kstest_and_ks1samp(x, 'less', mode='exact')
  
      # missing: no test that uses *args
  
  class TestKSOneSample:
      """Tests kstest and ks_samp 1-samples with K-S various sizes, alternatives, modes."""
  
      def _testOne(self, x, alternative, expected_statistic, expected_prob, mode='auto', decimal=14):
          result = stats.ks_1samp(x, stats.norm.cdf, alternative=alternative, mode=mode)
          expected = np.array([expected_statistic, expected_prob])
          assert_array_almost_equal(np.array(result), expected, decimal=decimal)
  
      def test_namedtuple_attributes(self):
          x = np.linspace(-1, 1, 9)
          # test for namedtuple attribute results
          attributes = ('statistic', 'pvalue')
          res = stats.ks_1samp(x, stats.norm.cdf)
          check_named_results(res, attributes)
  
      def test_agree_with_r(self):
          # comparing with some values from R
          x = np.linspace(-1, 1, 9)
          self._testOne(x, 'two-sided', 0.15865525393145705, 0.95164069201518386)
  
          x = np.linspace(-15, 15, 9)
          self._testOne(x, 'two-sided', 0.44435602715924361, 0.038850140086788665)
  
          x = [-1.23, 0.06, -0.60, 0.17, 0.66, -0.17, -0.08, 0.27, -0.98, -0.99]
          self._testOne(x, 'two-sided', 0.293580126801961, 0.293408463684361)
          self._testOne(x, 'greater', 0.293580126801961, 0.146988835042376, mode='exact')
          self._testOne(x, 'less', 0.109348552425692, 0.732768892470675, mode='exact')
  
      def test_known_examples(self):
          # the following tests rely on deterministically replicated rvs
          x = stats.norm.rvs(loc=0.2, size=100, random_state=987654321)
          self._testOne(x, 'two-sided', 0.12464329735846891, 0.089444888711820769, mode='asymp')
          self._testOne(x, 'less', 0.12464329735846891, 0.040989164077641749)
          self._testOne(x, 'greater', 0.0072115233216310994, 0.98531158590396228)
  
      def test_ks1samp_allpaths(self):
          # Check NaN input, output.
          assert_(np.isnan(kolmogn(np.nan, 1, True)))
          with assert_raises(ValueError, match='n is not integral: 1.5'):
              kolmogn(1.5, 1, True)
          assert_(np.isnan(kolmogn(-1, 1, True)))
  
          dataset = np.asarray([
              # Check x out of range
              (101, 1, True, 1.0),
              (101, 1.1, True, 1.0),
              (101, 0, True, 0.0),
              (101, -0.1, True, 0.0),
  
              (32, 1.0 / 64, True, 0.0),  # Ruben-Gambino
              (32, 1.0 / 64, False, 1.0),  # Ruben-Gambino
  
              (32, 0.5, True, 0.9999999363163307),  # Miller
              (32, 0.5, False, 6.368366937916623e-08),  # Miller 2 * special.smirnov(32, 0.5)
  
              # Check some other paths
              (32, 1.0 / 8, True, 0.34624229979775223),
              (32, 1.0 / 4, True, 0.9699508336558085),
              (1600, 0.49, False, 0.0),
              (1600, 1 / 16.0, False, 7.0837876229702195e-06),  # 2 * special.smirnov(1600, 1/16.0)
              (1600, 14 / 1600, False, 0.99962357317602),  # _kolmogn_DMTW
              (1600, 1 / 32, False, 0.08603386296651416),  # _kolmogn_PelzGood
          ])
          FuncData(kolmogn, dataset, (0, 1, 2), 3).check(dtypes=[int, float, bool])
  
      # missing: no test that uses *args
  
  
  class TestKSTwoSamples:
      """Tests 2-samples with K-S various sizes, alternatives, modes."""
  
      def _testOne(self, x1, x2, alternative, expected_statistic, expected_prob, mode='auto'):
          result = stats.ks_2samp(x1, x2, alternative, mode=mode)
          expected = np.array([expected_statistic, expected_prob])
          assert_array_almost_equal(np.array(result), expected)
  
      def testSmall(self):
          self._testOne([0], [1], 'two-sided', 1.0/1, 1.0)
          self._testOne([0], [1], 'greater', 1.0/1, 0.5)
          self._testOne([0], [1], 'less', 0.0/1, 1.0)
          self._testOne([1], [0], 'two-sided', 1.0/1, 1.0)
          self._testOne([1], [0], 'greater', 0.0/1, 1.0)
          self._testOne([1], [0], 'less', 1.0/1, 0.5)
  
      def testTwoVsThree(self):
          data1 = np.array([1.0, 2.0])
          data1p = data1 + 0.01
          data1m = data1 - 0.01
          data2 = np.array([1.0, 2.0, 3.0])
          self._testOne(data1p, data2, 'two-sided', 1.0 / 3, 1.0)
          self._testOne(data1p, data2, 'greater', 1.0 / 3, 0.7)
          self._testOne(data1p, data2, 'less', 1.0 / 3, 0.7)
          self._testOne(data1m, data2, 'two-sided', 2.0 / 3, 0.6)
          self._testOne(data1m, data2, 'greater', 2.0 / 3, 0.3)
          self._testOne(data1m, data2, 'less', 0, 1.0)
  
      def testTwoVsFour(self):
          data1 = np.array([1.0, 2.0])
          data1p = data1 + 0.01
          data1m = data1 - 0.01
          data2 = np.array([1.0, 2.0, 3.0, 4.0])
          self._testOne(data1p, data2, 'two-sided', 2.0 / 4, 14.0/15)
          self._testOne(data1p, data2, 'greater', 2.0 / 4, 8.0/15)
          self._testOne(data1p, data2, 'less', 1.0 / 4, 12.0/15)
  
          self._testOne(data1m, data2, 'two-sided', 3.0 / 4, 6.0/15)
          self._testOne(data1m, data2, 'greater', 3.0 / 4, 3.0/15)
          self._testOne(data1m, data2, 'less', 0, 1.0)
  
      def test100_100(self):
          x100 = np.linspace(1, 100, 100)
          x100_2_p1 = x100 + 2 + 0.1
          x100_2_m1 = x100 + 2 - 0.1
          self._testOne(x100, x100_2_p1, 'two-sided', 3.0 / 100, 0.9999999999962055)
          self._testOne(x100, x100_2_p1, 'greater', 3.0 / 100, 0.9143290114276248)
          self._testOne(x100, x100_2_p1, 'less', 0, 1.0)
          self._testOne(x100, x100_2_m1, 'two-sided', 2.0 / 100, 1.0)
          self._testOne(x100, x100_2_m1, 'greater', 2.0 / 100, 0.960978450786184)
          self._testOne(x100, x100_2_m1, 'less', 0, 1.0)
  
      def test100_110(self):
          x100 = np.linspace(1, 100, 100)
          x110 = np.linspace(1, 100, 110)
          x110_20_p1 = x110 + 20 + 0.1
          x110_20_m1 = x110 + 20 - 0.1
          # 100, 110
          self._testOne(x100, x110_20_p1, 'two-sided', 232.0 / 1100, 0.015739183865607353)
          self._testOne(x100, x110_20_p1, 'greater', 232.0 / 1100, 0.007869594319053203)
          self._testOne(x100, x110_20_p1, 'less', 0, 1)
          self._testOne(x100, x110_20_m1, 'two-sided', 229.0 / 1100, 0.017803803861026313)
          self._testOne(x100, x110_20_m1, 'greater', 229.0 / 1100, 0.008901905958245056)
          self._testOne(x100, x110_20_m1, 'less', 0.0, 1.0)
  
      def testRepeatedValues(self):
          x2233 = np.array([2] * 3 + [3] * 4 + [5] * 5 + [6] * 4, dtype=int)
          x3344 = x2233 + 1
          x2356 = np.array([2] * 3 + [3] * 4 + [5] * 10 + [6] * 4, dtype=int)
          x3467 = np.array([3] * 10 + [4] * 2 + [6] * 10 + [7] * 4, dtype=int)
          self._testOne(x2233, x3344, 'two-sided', 5.0/16, 0.4262934613454952)
          self._testOne(x2233, x3344, 'greater', 5.0/16, 0.21465428276573786)
          self._testOne(x2233, x3344, 'less', 0.0/16, 1.0)
          self._testOne(x2356, x3467, 'two-sided', 190.0/21/26, 0.0919245790168125)
          self._testOne(x2356, x3467, 'greater', 190.0/21/26, 0.0459633806858544)
          self._testOne(x2356, x3467, 'less', 70.0/21/26, 0.6121593130022775)
  
      def testEqualSizes(self):
          data2 = np.array([1.0, 2.0, 3.0])
          self._testOne(data2, data2+1, 'two-sided', 1.0/3, 1.0)
          self._testOne(data2, data2+1, 'greater', 1.0/3, 0.75)
          self._testOne(data2, data2+1, 'less', 0.0/3, 1.)
          self._testOne(data2, data2+0.5, 'two-sided', 1.0/3, 1.0)
          self._testOne(data2, data2+0.5, 'greater', 1.0/3, 0.75)
          self._testOne(data2, data2+0.5, 'less', 0.0/3, 1.)
          self._testOne(data2, data2-0.5, 'two-sided', 1.0/3, 1.0)
          self._testOne(data2, data2-0.5, 'greater', 0.0/3, 1.0)
          self._testOne(data2, data2-0.5, 'less', 1.0/3, 0.75)
  
      @pytest.mark.slow
      def testMiddlingBoth(self):
          # 500, 600
          n1, n2 = 500, 600
          delta = 1.0/n1/n2/2/2
          x = np.linspace(1, 200, n1) - delta
          y = np.linspace(2, 200, n2)
          self._testOne(x, y, 'two-sided', 2000.0 / n1 / n2, 1.0, mode='auto')
          self._testOne(x, y, 'two-sided', 2000.0 / n1 / n2, 1.0, mode='asymp')
          self._testOne(x, y, 'greater', 2000.0 / n1 / n2, 0.9697596024683929, mode='asymp')
          self._testOne(x, y, 'less', 500.0 / n1 / n2, 0.9968735843165021, mode='asymp')
          with suppress_warnings() as sup:
              message = "ks_2samp: Exact calculation unsuccessful."
              sup.filter(RuntimeWarning, message)
              self._testOne(x, y, 'greater', 2000.0 / n1 / n2, 0.9697596024683929, mode='exact')
              self._testOne(x, y, 'less', 500.0 / n1 / n2, 0.9968735843165021, mode='exact')
          with warnings.catch_warnings(record=True) as w:
              warnings.simplefilter("always")
              self._testOne(x, y, 'less', 500.0 / n1 / n2, 0.9968735843165021, mode='exact')
              _check_warnings(w, RuntimeWarning, 1)
  
      @pytest.mark.slow
      def testMediumBoth(self):
          # 1000, 1100
          n1, n2 = 1000, 1100
          delta = 1.0/n1/n2/2/2
          x = np.linspace(1, 200, n1) - delta
          y = np.linspace(2, 200, n2)
          self._testOne(x, y, 'two-sided', 6600.0 / n1 / n2, 1.0, mode='asymp')
          self._testOne(x, y, 'two-sided', 6600.0 / n1 / n2, 1.0, mode='auto')
          self._testOne(x, y, 'greater', 6600.0 / n1 / n2, 0.9573185808092622, mode='asymp')
          self._testOne(x, y, 'less', 1000.0 / n1 / n2, 0.9982410869433984, mode='asymp')
  
          with suppress_warnings() as sup:
              message = "ks_2samp: Exact calculation unsuccessful."
              sup.filter(RuntimeWarning, message)
              self._testOne(x, y, 'greater', 6600.0 / n1 / n2, 0.9573185808092622, mode='exact')
              self._testOne(x, y, 'less', 1000.0 / n1 / n2, 0.9982410869433984, mode='exact')
          with warnings.catch_warnings(record=True) as w:
              warnings.simplefilter("always")
              self._testOne(x, y, 'less', 1000.0 / n1 / n2, 0.9982410869433984, mode='exact')
              _check_warnings(w, RuntimeWarning, 1)
  
      def testLarge(self):
          # 10000, 110
          n1, n2 = 10000, 110
          lcm = n1*11.0
          delta = 1.0/n1/n2/2/2
          x = np.linspace(1, 200, n1) - delta
          y = np.linspace(2, 100, n2)
          self._testOne(x, y, 'two-sided', 55275.0 / lcm, 4.2188474935755949e-15)
          self._testOne(x, y, 'greater', 561.0 / lcm, 0.99115454582047591)
          self._testOne(x, y, 'less', 55275.0 / lcm, 3.1317328311518713e-26)
  
      def test_gh11184(self):
          # 3000, 3001, exact two-sided
          np.random.seed(123456)
          x = np.random.normal(size=3000)
          y = np.random.normal(size=3001) * 1.5
          self._testOne(x, y, 'two-sided', 0.11292880151060758, 2.7755575615628914e-15, mode='asymp')
          self._testOne(x, y, 'two-sided', 0.11292880151060758, 2.7755575615628914e-15, mode='exact')
  
      @pytest.mark.xslow
      def test_gh11184_bigger(self):
          # 10000, 10001, exact two-sided
          np.random.seed(123456)
          x = np.random.normal(size=10000)
          y = np.random.normal(size=10001) * 1.5
          self._testOne(x, y, 'two-sided', 0.10597913208679133, 3.3149311398483503e-49, mode='asymp')
          self._testOne(x, y, 'two-sided', 0.10597913208679133, 2.7755575615628914e-15, mode='exact')
          self._testOne(x, y, 'greater', 0.10597913208679133, 2.7947433906389253e-41, mode='asymp')
          self._testOne(x, y, 'less', 0.09658002199780022, 2.7947433906389253e-41, mode='asymp')
  
      @pytest.mark.xslow
      def test_gh12999(self):
          np.random.seed(123456)
          for x in range(1000, 12000, 1000):
              vals1 = np.random.normal(size=(x))
              vals2 = np.random.normal(size=(x + 10), loc=0.5)
              exact = stats.ks_2samp(vals1, vals2, mode='exact').pvalue
              asymp = stats.ks_2samp(vals1, vals2, mode='asymp').pvalue
              # these two p-values should be in line with each other
              assert_array_less(exact, 3 * asymp)
              assert_array_less(asymp, 3 * exact)
  
      @pytest.mark.slow
      def testLargeBoth(self):
          # 10000, 11000
          n1, n2 = 10000, 11000
          lcm = n1*11.0
          delta = 1.0/n1/n2/2/2
          x = np.linspace(1, 200, n1) - delta
          y = np.linspace(2, 200, n2)
          self._testOne(x, y, 'two-sided', 563.0 / lcm, 0.9990660108966576, mode='asymp')
          self._testOne(x, y, 'two-sided', 563.0 / lcm, 0.9990456491488628, mode='exact')
          self._testOne(x, y, 'two-sided', 563.0 / lcm, 0.9990660108966576, mode='auto')
          self._testOne(x, y, 'greater', 563.0 / lcm, 0.7561851877420673)
          self._testOne(x, y, 'less', 10.0 / lcm, 0.9998239693191724)
          with suppress_warnings() as sup:
              message = "ks_2samp: Exact calculation unsuccessful."
              sup.filter(RuntimeWarning, message)
              self._testOne(x, y, 'greater', 563.0 / lcm, 0.7561851877420673, mode='exact')
              self._testOne(x, y, 'less', 10.0 / lcm, 0.9998239693191724, mode='exact')
  
      def testNamedAttributes(self):
          # test for namedtuple attribute results
          attributes = ('statistic', 'pvalue')
          res = stats.ks_2samp([1, 2], [3])
          check_named_results(res, attributes)
  
      @pytest.mark.slow
      def test_some_code_paths(self):
          # Check that some code paths are executed
          from scipy.stats._stats_py import (
              _count_paths_outside_method,
              _compute_outer_prob_inside_method
          )
  
          _compute_outer_prob_inside_method(1, 1, 1, 1)
          _count_paths_outside_method(1000, 1, 1, 1001)
  
          assert_raises(FloatingPointError, _count_paths_outside_method, 1100, 1099, 1, 1)
          assert_raises(FloatingPointError, _count_paths_outside_method, 2000, 1000, 1, 1)
  
      def test_argument_checking(self):
          # Check that an empty array causes a ValueError
          assert_raises(ValueError, stats.ks_2samp, [], [1])
          assert_raises(ValueError, stats.ks_2samp, [1], [])
          assert_raises(ValueError, stats.ks_2samp, [], [])
  
      @pytest.mark.slow
      def test_gh12218(self):
          """Ensure gh-12218 is fixed."""
          # gh-1228 triggered a TypeError calculating sqrt(n1*n2*(n1+n2)).
          # n1, n2 both large integers, the product exceeded 2^64
          np.random.seed(12345678)
          n1 = 2097152  # 2*^21
          rvs1 = stats.uniform.rvs(size=n1, loc=0., scale=1)
          rvs2 = rvs1 + 1  # Exact value of rvs2 doesn't matter.
          stats.ks_2samp(rvs1, rvs2, alternative='greater', mode='asymp')
          stats.ks_2samp(rvs1, rvs2, alternative='less', mode='asymp')
          stats.ks_2samp(rvs1, rvs2, alternative='two-sided', mode='asymp')
  
  
  def test_ttest_rel():
      # regression test
      tr,pr = 0.81248591389165692, 0.41846234511362157
      tpr = ([tr,-tr],[pr,pr])
  
      rvs1 = np.linspace(1,100,100)
      rvs2 = np.linspace(1.01,99.989,100)
      rvs1_2D = np.array([np.linspace(1,100,100), np.linspace(1.01,99.989,100)])
      rvs2_2D = np.array([np.linspace(1.01,99.989,100), np.linspace(1,100,100)])
  
      t,p = stats.ttest_rel(rvs1, rvs2, axis=0)
      assert_array_almost_equal([t,p],(tr,pr))
      t,p = stats.ttest_rel(rvs1_2D.T, rvs2_2D.T, axis=0)
      assert_array_almost_equal([t,p],tpr)
      t,p = stats.ttest_rel(rvs1_2D, rvs2_2D, axis=1)
      assert_array_almost_equal([t,p],tpr)
  
      # test scalars
      with suppress_warnings() as sup, np.errstate(invalid="ignore"), \
              pytest.warns(RuntimeWarning, match="Precision loss occurred"):
          sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
          t, p = stats.ttest_rel(4., 3.)
      assert_(np.isnan(t))
      assert_(np.isnan(p))
  
      # test for namedtuple attribute results
      attributes = ('statistic', 'pvalue')
      res = stats.ttest_rel(rvs1, rvs2, axis=0)
      check_named_results(res, attributes)
  
      # test on 3 dimensions
      rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
      rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
      t,p = stats.ttest_rel(rvs1_3D, rvs2_3D, axis=1)
      assert_array_almost_equal(np.abs(t), tr)
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (2, 3))
  
      t, p = stats.ttest_rel(np.moveaxis(rvs1_3D, 2, 0),
                             np.moveaxis(rvs2_3D, 2, 0),
                             axis=2)
      assert_array_almost_equal(np.abs(t), tr)
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (3, 2))
  
      # test alternative parameter
      assert_raises(ValueError, stats.ttest_rel, rvs1, rvs2, alternative="error")
  
      t, p = stats.ttest_rel(rvs1, rvs2, axis=0, alternative="less")
      assert_allclose(p, 1 - pr/2)
      assert_allclose(t, tr)
  
      t, p = stats.ttest_rel(rvs1, rvs2, axis=0, alternative="greater")
      assert_allclose(p, pr/2)
      assert_allclose(t, tr)
  
      # check nan policy
      rng = np.random.RandomState(12345678)
      x = stats.norm.rvs(loc=5, scale=10, size=501, random_state=rng)
      x[500] = np.nan
      y = (stats.norm.rvs(loc=5, scale=10, size=501, random_state=rng) +
           stats.norm.rvs(scale=0.2, size=501, random_state=rng))
      y[500] = np.nan
  
      with np.errstate(invalid="ignore"):
          assert_array_equal(stats.ttest_rel(x, x), (np.nan, np.nan))
  
      assert_array_almost_equal(stats.ttest_rel(x, y, nan_policy='omit'),
                                (0.25299925303978066, 0.8003729814201519))
      assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='raise')
      assert_raises(ValueError, stats.ttest_rel, x, y, nan_policy='foobar')
  
      # test zero division problem
      with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
          t, p = stats.ttest_rel([0, 0, 0], [1, 1, 1])
      assert_equal((np.abs(t), p), (np.inf, 0))
      with np.errstate(invalid="ignore"):
          assert_equal(stats.ttest_rel([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))
  
          # check that nan in input array result in nan output
          anan = np.array([[1, np.nan], [-1, 1]])
          assert_equal(stats.ttest_rel(anan, np.zeros((2, 2))),
                       ([0, np.nan], [1, np.nan]))
  
      # test incorrect input shape raise an error
      x = np.arange(24)
      assert_raises(ValueError, stats.ttest_rel, x.reshape((8, 3)),
                    x.reshape((2, 3, 4)))
  
      # Convert from two-sided p-values to one sided using T result data.
      def convert(t, p, alt):
          if (t < 0 and alt == "less") or (t > 0 and alt == "greater"):
              return p / 2
          return 1 - (p / 2)
      converter = np.vectorize(convert)
  
      rvs1_2D[:, 20:30] = np.nan
      rvs2_2D[:, 15:25] = np.nan
  
      tr, pr = stats.ttest_rel(rvs1_2D, rvs2_2D, 0, nan_policy='omit')
  
      t, p = stats.ttest_rel(rvs1_2D, rvs2_2D, 0, nan_policy='omit',
                             alternative='less')
      assert_allclose(t, tr, rtol=1e-14)
      assert_allclose(p, converter(tr, pr, 'less'), rtol=1e-14)
  
      t, p = stats.ttest_rel(rvs1_2D, rvs2_2D, 0, nan_policy='omit',
                             alternative='greater')
      assert_allclose(t, tr, rtol=1e-14)
      assert_allclose(p, converter(tr, pr, 'greater'), rtol=1e-14)
  
  
  def test_ttest_rel_nan_2nd_arg():
      # regression test for gh-6134: nans in the second arg were not handled
      x = [np.nan, 2.0, 3.0, 4.0]
      y = [1.0, 2.0, 1.0, 2.0]
  
      r1 = stats.ttest_rel(x, y, nan_policy='omit')
      r2 = stats.ttest_rel(y, x, nan_policy='omit')
      assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
      assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)
  
      # NB: arguments are paired when NaNs are dropped
      r3 = stats.ttest_rel(y[1:], x[1:])
      assert_allclose(r2, r3, atol=1e-15)
  
      # .. and this is consistent with R. R code:
      # x = c(NA, 2.0, 3.0, 4.0)
      # y = c(1.0, 2.0, 1.0, 2.0)
      # t.test(x, y, paired=TRUE)
      assert_allclose(r2, (-2, 0.1835), atol=1e-4)
  
  
  def test_ttest_rel_empty_1d_returns_nan():
      # Two empty inputs should return a Ttest_relResult containing nan
      # for both values.
      result = stats.ttest_rel([], [])
      assert isinstance(result, stats._stats_py.Ttest_relResult)
      assert_equal(result, (np.nan, np.nan))
  
  
  @pytest.mark.parametrize('b, expected_shape',
                           [(np.empty((1, 5, 0)), (3, 5)),
                            (np.empty((1, 0, 0)), (3, 0))])
  def test_ttest_rel_axis_size_zero(b, expected_shape):
      # In this test, the length of the axis dimension is zero.
      # The results should be arrays containing nan with shape
      # given by the broadcast nonaxis dimensions.
      a = np.empty((3, 1, 0))
      result = stats.ttest_rel(a, b, axis=-1)
      assert isinstance(result, stats._stats_py.Ttest_relResult)
      expected_value = np.full(expected_shape, fill_value=np.nan)
      assert_equal(result.statistic, expected_value)
      assert_equal(result.pvalue, expected_value)
  
  
  def test_ttest_rel_nonaxis_size_zero():
      # In this test, the length of the axis dimension is nonzero,
      # but one of the nonaxis dimensions has length 0.  Check that
      # we still get the correctly broadcast shape, which is (5, 0)
      # in this case.
      a = np.empty((1, 8, 0))
      b = np.empty((5, 8, 1))
      result = stats.ttest_rel(a, b, axis=1)
      assert isinstance(result, stats._stats_py.Ttest_relResult)
      assert_equal(result.statistic.shape, (5, 0))
      assert_equal(result.pvalue.shape, (5, 0))
  
  
  def _desc_stats(x1, x2, axis=0):
      def _stats(x, axis=0):
          x = np.asarray(x)
          mu = np.mean(x, axis=axis)
          std = np.std(x, axis=axis, ddof=1)
          nobs = x.shape[axis]
          return mu, std, nobs
      return _stats(x1, axis) + _stats(x2, axis)
  
  
  def test_ttest_ind():
      # regression test
      tr = 1.0912746897927283
      pr = 0.27647818616351882
      tpr = ([tr,-tr],[pr,pr])
  
      rvs2 = np.linspace(1,100,100)
      rvs1 = np.linspace(5,105,100)
      rvs1_2D = np.array([rvs1, rvs2])
      rvs2_2D = np.array([rvs2, rvs1])
  
      t,p = stats.ttest_ind(rvs1, rvs2, axis=0)
      assert_array_almost_equal([t,p],(tr,pr))
      # test from_stats API
      assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
                                                                        rvs2)),
                                [t, p])
      t,p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0)
      assert_array_almost_equal([t,p],tpr)
      args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
      assert_array_almost_equal(stats.ttest_ind_from_stats(*args),
                                [t, p])
      t,p = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1)
      assert_array_almost_equal([t,p],tpr)
      args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
      assert_array_almost_equal(stats.ttest_ind_from_stats(*args),
                                [t, p])
  
      # test scalars
      with suppress_warnings() as sup, np.errstate(invalid="ignore"), \
              pytest.warns(RuntimeWarning, match="Precision loss occurred"):
          sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
          t, p = stats.ttest_ind(4., 3.)
      assert_(np.isnan(t))
      assert_(np.isnan(p))
  
      # test on 3 dimensions
      rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
      rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
      t,p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=1)
      assert_almost_equal(np.abs(t), np.abs(tr))
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (2, 3))
  
      t, p = stats.ttest_ind(np.moveaxis(rvs1_3D, 2, 0),
                             np.moveaxis(rvs2_3D, 2, 0),
                             axis=2)
      assert_array_almost_equal(np.abs(t), np.abs(tr))
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (3, 2))
  
      # test alternative parameter
      assert_raises(ValueError, stats.ttest_ind, rvs1, rvs2, alternative="error")
      assert_raises(ValueError, stats.ttest_ind_from_stats,
                    *_desc_stats(rvs1_2D.T, rvs2_2D.T), alternative="error")
  
      t, p = stats.ttest_ind(rvs1, rvs2, alternative="less")
      assert_allclose(p, 1 - (pr/2))
      assert_allclose(t, tr)
  
      t, p = stats.ttest_ind(rvs1, rvs2, alternative="greater")
      assert_allclose(p, pr/2)
      assert_allclose(t, tr)
  
      # Below makes sure ttest_ind_from_stats p-val functions identically to
      # ttest_ind
      t, p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, alternative="less")
      args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
      assert_allclose(
          stats.ttest_ind_from_stats(*args, alternative="less"), [t, p])
  
      t, p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, alternative="greater")
      args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
      assert_allclose(
          stats.ttest_ind_from_stats(*args, alternative="greater"), [t, p])
  
      # check nan policy
      rng = np.random.RandomState(12345678)
      x = stats.norm.rvs(loc=5, scale=10, size=501, random_state=rng)
      x[500] = np.nan
      y = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng)
  
      with np.errstate(invalid="ignore"):
          assert_array_equal(stats.ttest_ind(x, y), (np.nan, np.nan))
  
      assert_array_almost_equal(stats.ttest_ind(x, y, nan_policy='omit'),
                                (0.24779670949091914, 0.80434267337517906))
      assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='raise')
      assert_raises(ValueError, stats.ttest_ind, x, y, nan_policy='foobar')
  
      # test zero division problem
      with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
          t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1])
      assert_equal((np.abs(t), p), (np.inf, 0))
  
      with np.errstate(invalid="ignore"):
          assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0]), (np.nan, np.nan))
  
          # check that nan in input array result in nan output
          anan = np.array([[1, np.nan], [-1, 1]])
          assert_equal(stats.ttest_ind(anan, np.zeros((2, 2))),
                       ([0, np.nan], [1, np.nan]))
  
      rvs1_3D[:, :, 10:15] = np.nan
      rvs2_3D[:, :, 6:12] = np.nan
  
      # Convert from two-sided p-values to one sided using T result data.
      def convert(t, p, alt):
          if (t < 0 and alt == "less") or (t > 0 and alt == "greater"):
              return p / 2
          return 1 - (p / 2)
      converter = np.vectorize(convert)
  
      tr, pr = stats.ttest_ind(rvs1_3D, rvs2_3D, 0, nan_policy='omit')
  
      t, p = stats.ttest_ind(rvs1_3D, rvs2_3D, 0, nan_policy='omit',
                          alternative='less')
      assert_allclose(t, tr, rtol=1e-14)
      assert_allclose(p, converter(tr, pr, 'less'), rtol=1e-14)
  
      t, p = stats.ttest_ind(rvs1_3D, rvs2_3D, 0, nan_policy='omit',
                          alternative='greater')
      assert_allclose(t, tr, rtol=1e-14)
      assert_allclose(p, converter(tr, pr, 'greater'), rtol=1e-14)
  
  
  class Test_ttest_ind_permutations():
      N = 20
  
      # data for most tests
      np.random.seed(0)
      a = np.vstack((np.arange(3*N//4), np.random.random(3*N//4)))
      b = np.vstack((np.arange(N//4) + 100, np.random.random(N//4)))
  
      # data for equal variance tests
      a2 = np.arange(10)
      b2 = np.arange(10) + 100
  
      # data for exact test
      a3 = [1, 2]
      b3 = [3, 4]
  
      # data for bigger test
      np.random.seed(0)
      rvs1 = stats.norm.rvs(loc=5, scale=10,  # type: ignore
                            size=500).reshape(100, 5).T
      rvs2 = stats.norm.rvs(loc=8, scale=20, size=100)  # type: ignore
  
      p_d = [0, 0.676]  # desired pvalues
      p_d_gen = [0, 0.672]  # desired pvalues for Generator seed
      p_d_big = [0.993, 0.685, 0.84, 0.955, 0.255]
  
      params = [
          (a, b, {"axis": 1}, p_d),                     # basic test
          (a.T, b.T, {'axis': 0}, p_d),                 # along axis 0
          (a[0, :], b[0, :], {'axis': None}, p_d[0]),   # 1d data
          (a[0, :].tolist(), b[0, :].tolist(), {'axis': None}, p_d[0]),
          # different seeds
          (a, b, {'random_state': 0, "axis": 1}, p_d),
          (a, b, {'random_state': np.random.RandomState(0), "axis": 1}, p_d),
          (a2, b2, {'equal_var': True}, 0),  # equal variances
          (rvs1, rvs2, {'axis': -1, 'random_state': 0}, p_d_big),  # bigger test
          (a3, b3, {}, 1/3)  # exact test
          ]
  
      if NumpyVersion(np.__version__) >= '1.18.0':
          params.append(
              (a, b, {'random_state': np.random.default_rng(0), "axis": 1},
               p_d_gen),
              )
  
      @pytest.mark.parametrize("a,b,update,p_d", params)
      def test_ttest_ind_permutations(self, a, b, update, p_d):
          options_a = {'axis': None, 'equal_var': False}
          options_p = {'axis': None, 'equal_var': False,
                       'permutations': 1000, 'random_state': 0}
          options_a.update(update)
          options_p.update(update)
  
          stat_a, _ = stats.ttest_ind(a, b, **options_a)
          stat_p, pvalue = stats.ttest_ind(a, b, **options_p)
          assert_array_almost_equal(stat_a, stat_p, 5)
          assert_array_almost_equal(pvalue, p_d)
  
      def test_ttest_ind_exact_alternative(self):
          np.random.seed(0)
          N = 3
          a = np.random.rand(2, N, 2)
          b = np.random.rand(2, N, 2)
  
          options_p = {'axis': 1, 'permutations': 1000}
  
          options_p.update(alternative="greater")
          res_g_ab = stats.ttest_ind(a, b, **options_p)
          res_g_ba = stats.ttest_ind(b, a, **options_p)
  
          options_p.update(alternative="less")
          res_l_ab = stats.ttest_ind(a, b, **options_p)
          res_l_ba = stats.ttest_ind(b, a, **options_p)
  
          options_p.update(alternative="two-sided")
          res_2_ab = stats.ttest_ind(a, b, **options_p)
          res_2_ba = stats.ttest_ind(b, a, **options_p)
  
          # Alternative doesn't affect the statistic
          assert_equal(res_g_ab.statistic, res_l_ab.statistic)
          assert_equal(res_g_ab.statistic, res_2_ab.statistic)
  
          # Reversing order of inputs negates statistic
          assert_equal(res_g_ab.statistic, -res_g_ba.statistic)
          assert_equal(res_l_ab.statistic, -res_l_ba.statistic)
          assert_equal(res_2_ab.statistic, -res_2_ba.statistic)
  
          # Reversing order of inputs does not affect p-value of 2-sided test
          assert_equal(res_2_ab.pvalue, res_2_ba.pvalue)
  
          # In exact test, distribution is perfectly symmetric, so these
          # identities are exactly satisfied.
          assert_equal(res_g_ab.pvalue, res_l_ba.pvalue)
          assert_equal(res_l_ab.pvalue, res_g_ba.pvalue)
          mask = res_g_ab.pvalue <= 0.5
          assert_equal(res_g_ab.pvalue[mask] + res_l_ba.pvalue[mask],
                       res_2_ab.pvalue[mask])
          assert_equal(res_l_ab.pvalue[~mask] + res_g_ba.pvalue[~mask],
                       res_2_ab.pvalue[~mask])
  
      def test_ttest_ind_exact_selection(self):
          # test the various ways of activating the exact test
          np.random.seed(0)
          N = 3
          a = np.random.rand(N)
          b = np.random.rand(N)
          res0 = stats.ttest_ind(a, b)
          res1 = stats.ttest_ind(a, b, permutations=1000)
          res2 = stats.ttest_ind(a, b, permutations=0)
          res3 = stats.ttest_ind(a, b, permutations=np.inf)
          assert(res1.pvalue != res0.pvalue)
          assert(res2.pvalue == res0.pvalue)
          assert(res3.pvalue == res1.pvalue)
  
      def test_ttest_ind_exact_distribution(self):
          # the exact distribution of the test statistic should have
          # binom(na + nb, na) elements, all unique. This was not always true
          # in gh-4824; fixed by gh-13661.
          np.random.seed(0)
          a = np.random.rand(3)
          b = np.random.rand(4)
  
          data = np.concatenate((a, b))
          na, nb = len(a), len(b)
  
          permutations = 100000
          t_stat, _ = _permutation_distribution_t(data, permutations, na, True)
  
          n_unique = len(set(t_stat))
          assert n_unique == binom(na + nb, na)
          assert len(t_stat) == n_unique
  
      def test_ttest_ind_randperm_alternative(self):
          np.random.seed(0)
          N = 50
          a = np.random.rand(2, 3, N)
          b = np.random.rand(3, N)
          options_p = {'axis': -1, 'permutations': 1000, "random_state": 0}
  
          options_p.update(alternative="greater")
          res_g_ab = stats.ttest_ind(a, b, **options_p)
          res_g_ba = stats.ttest_ind(b, a, **options_p)
  
          options_p.update(alternative="less")
          res_l_ab = stats.ttest_ind(a, b, **options_p)
          res_l_ba = stats.ttest_ind(b, a, **options_p)
  
          # Alternative doesn't affect the statistic
          assert_equal(res_g_ab.statistic, res_l_ab.statistic)
  
          # Reversing order of inputs negates statistic
          assert_equal(res_g_ab.statistic, -res_g_ba.statistic)
          assert_equal(res_l_ab.statistic, -res_l_ba.statistic)
  
          # For random permutations, the chance of ties between the observed
          # test statistic and the population is small, so:
          assert_equal(res_g_ab.pvalue + res_l_ab.pvalue, 1)
          assert_equal(res_g_ba.pvalue + res_l_ba.pvalue, 1)
  
      @pytest.mark.slow()
      def test_ttest_ind_randperm_alternative2(self):
          np.random.seed(0)
          N = 50
          a = np.random.rand(N, 4)
          b = np.random.rand(N, 4)
          options_p = {'permutations': 20000, "random_state": 0}
  
          options_p.update(alternative="greater")
          res_g_ab = stats.ttest_ind(a, b, **options_p)
  
          options_p.update(alternative="less")
          res_l_ab = stats.ttest_ind(a, b, **options_p)
  
          options_p.update(alternative="two-sided")
          res_2_ab = stats.ttest_ind(a, b, **options_p)
  
          # For random permutations, the chance of ties between the observed
          # test statistic and the population is small, so:
          assert_equal(res_g_ab.pvalue + res_l_ab.pvalue, 1)
  
          # For for large sample sizes, the distribution should be approximately
          # symmetric, so these identities should be approximately satisfied
          mask = res_g_ab.pvalue <= 0.5
          assert_allclose(2 * res_g_ab.pvalue[mask],
                          res_2_ab.pvalue[mask], atol=2e-2)
          assert_allclose(2 * (1-res_g_ab.pvalue[~mask]),
                          res_2_ab.pvalue[~mask], atol=2e-2)
          assert_allclose(2 * res_l_ab.pvalue[~mask],
                          res_2_ab.pvalue[~mask], atol=2e-2)
          assert_allclose(2 * (1-res_l_ab.pvalue[mask]),
                          res_2_ab.pvalue[mask], atol=2e-2)
  
      def test_ttest_ind_permutation_nanpolicy(self):
          np.random.seed(0)
          N = 50
          a = np.random.rand(N, 5)
          b = np.random.rand(N, 5)
          a[5, 1] = np.nan
          b[8, 2] = np.nan
          a[9, 3] = np.nan
          b[9, 3] = np.nan
          options_p = {'permutations': 1000, "random_state": 0}
  
          # Raise
          options_p.update(nan_policy="raise")
          with assert_raises(ValueError, match="The input contains nan values"):
              res = stats.ttest_ind(a, b, **options_p)
  
          # Propagate
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "invalid value*")
              options_p.update(nan_policy="propagate")
              res = stats.ttest_ind(a, b, **options_p)
  
              mask = np.isnan(a).any(axis=0) | np.isnan(b).any(axis=0)
              res2 = stats.ttest_ind(a[:, ~mask], b[:, ~mask], **options_p)
  
              assert_equal(res.pvalue[mask], np.nan)
              assert_equal(res.statistic[mask], np.nan)
  
              assert_allclose(res.pvalue[~mask], res2.pvalue)
              assert_allclose(res.statistic[~mask], res2.statistic)
  
              # Propagate 1d
              res = stats.ttest_ind(a.ravel(), b.ravel(), **options_p)
              assert(np.isnan(res.pvalue))  # assert makes sure it's a scalar
              assert(np.isnan(res.statistic))
  
      def test_ttest_ind_permutation_check_inputs(self):
          with assert_raises(ValueError, match="Permutations must be"):
              stats.ttest_ind(self.a2, self.b2, permutations=-3)
          with assert_raises(ValueError, match="Permutations must be"):
              stats.ttest_ind(self.a2, self.b2, permutations=1.5)
          with assert_raises(ValueError, match="'hello' cannot be used"):
              stats.ttest_ind(self.a, self.b, permutations=1,
                              random_state='hello')
  
  
  class Test_ttest_ind_common:
      # for tests that are performed on variations of the t-test such as
      # permutations and trimming
      @pytest.mark.slow()
      @pytest.mark.parametrize("kwds", [{'permutations': 200, 'random_state': 0},
                                        {'trim': .2}, {}],
                               ids=["permutations", "trim", "basic"])
      @pytest.mark.parametrize('equal_var', [True, False],
                               ids=['equal_var', 'unequal_var'])
      def test_ttest_many_dims(self, kwds, equal_var):
          # Test that test works on many-dimensional arrays
          np.random.seed(0)
          a = np.random.rand(5, 4, 4, 7, 1, 6)
          b = np.random.rand(4, 1, 8, 2, 6)
          res = stats.ttest_ind(a, b, axis=-3, **kwds)
  
          # compare fully-vectorized t-test against t-test on smaller slice
          i, j, k = 2, 3, 1
          a2 = a[i, :, j, :, 0, :]
          b2 = b[:, 0, :, k, :]
          res2 = stats.ttest_ind(a2, b2, axis=-2, **kwds)
          assert_equal(res.statistic[i, :, j, k, :],
                       res2.statistic)
          assert_equal(res.pvalue[i, :, j, k, :],
                       res2.pvalue)
  
          # compare against t-test on one axis-slice at a time
  
          # manually broadcast with tile; move axis to end to simplify
          x = np.moveaxis(np.tile(a, (1, 1, 1, 1, 2, 1)), -3, -1)
          y = np.moveaxis(np.tile(b, (5, 1, 4, 1, 1, 1)), -3, -1)
          shape = x.shape[:-1]
          statistics = np.zeros(shape)
          pvalues = np.zeros(shape)
          for indices in product(*(range(i) for i in shape)):
              xi = x[indices]  # use tuple to index single axis slice
              yi = y[indices]
              res3 = stats.ttest_ind(xi, yi, axis=-1, **kwds)
              statistics[indices] = res3.statistic
              pvalues[indices] = res3.pvalue
  
          assert_allclose(statistics, res.statistic)
          assert_allclose(pvalues, res.pvalue)
  
      @pytest.mark.parametrize("kwds", [{'permutations': 200, 'random_state': 0},
                                        {'trim': .2}, {}],
                               ids=["trim", "permutations", "basic"])
      @pytest.mark.parametrize("axis", [-1, 0])
      def test_nans_on_axis(self, kwds, axis):
          # confirm that with `nan_policy='propagate'`, NaN results are returned
          # on the correct location
          a = np.random.randint(10, size=(5, 3, 10)).astype('float')
          b = np.random.randint(10, size=(5, 3, 10)).astype('float')
          # set some indices in `a` and `b` to be `np.nan`.
          a[0][2][3] = np.nan
          b[2][0][6] = np.nan
  
          # arbitrarily use `np.sum` as a baseline for which indices should be
          # NaNs
          expected = np.isnan(np.sum(a + b, axis=axis))
          # multidimensional inputs to `t.sf(np.abs(t), df)` with NaNs on some
          # indices throws an warning. See issue gh-13844
          with suppress_warnings() as sup, np.errstate(invalid="ignore"):
              sup.filter(RuntimeWarning,
                         "invalid value encountered in less_equal")
              sup.filter(RuntimeWarning, "Precision loss occurred")
              res = stats.ttest_ind(a, b, axis=axis, **kwds)
          p_nans = np.isnan(res.pvalue)
          assert_array_equal(p_nans, expected)
          statistic_nans = np.isnan(res.statistic)
          assert_array_equal(statistic_nans, expected)
  
  
  class Test_ttest_trim:
      params = [
          [[1, 2, 3], [1.1, 2.9, 4.2], 0.53619490753126731, -0.6864951273557258,
           .2],
          [[56, 128.6, 12, 123.8, 64.34, 78, 763.3], [1.1, 2.9, 4.2],
           0.00998909252078421, 4.591598691181999, .2],
          [[56, 128.6, 12, 123.8, 64.34, 78, 763.3], [1.1, 2.9, 4.2],
           0.10512380092302633, 2.832256715395378, .32],
          [[2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9],
           [6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1],
           0.002878909511344, -4.2461168970325, .2],
          [[-0.84504783, 0.13366078, 3.53601757, -0.62908581, 0.54119466,
            -1.16511574, -0.08836614, 1.18495416, 2.48028757, -1.58925028,
            -1.6706357, 0.3090472, -2.12258305, 0.3697304, -1.0415207,
            -0.57783497, -0.90997008, 1.09850192, 0.41270579, -1.4927376],
           [1.2725522, 1.1657899, 2.7509041, 1.2389013, -0.9490494, -1.0752459,
            1.1038576, 2.9912821, 3.5349111, 0.4171922, 1.0168959, -0.7625041,
            -0.4300008, 3.0431921, 1.6035947, 0.5285634, -0.7649405, 1.5575896,
            1.3670797, 1.1726023], 0.005293305834235, -3.0983317739483, .2]]
  
      @pytest.mark.parametrize("a,b,pr,tr,trim", params)
      def test_ttest_compare_r(self, a, b, pr, tr, trim):
          '''
          Using PairedData's yuen.t.test method. Something to note is that there
          are at least 3 R packages that come with a trimmed t-test method, and
          comparisons were made between them. It was found that PairedData's
          method's results match this method, SAS, and one of the other R
          methods. A notable discrepancy was the DescTools implementation of the
          function, which only sometimes agreed with SAS, WRS2, PairedData and
          this implementation. For this reason, most comparisons in R are made
          against PairedData's method.
  
          Rather than providing the input and output for all evaluations, here is
          a representative example:
          > library(PairedData)
          > a <- c(1, 2, 3)
          > b <- c(1.1, 2.9, 4.2)
          > options(digits=16)
          > yuen.t.test(a, b, tr=.2)
  
              Two-sample Yuen test, trim=0.2
  
          data:  x and y
          t = -0.68649512735573, df = 3.4104431643464, p-value = 0.5361949075313
          alternative hypothesis: true difference in trimmed means is not equal
          to 0
          95 percent confidence interval:
           -3.912777195645217  2.446110528978550
          sample estimates:
          trimmed mean of x trimmed mean of y
          2.000000000000000 2.73333333333333
          '''
          statistic, pvalue = stats.ttest_ind(a, b, trim=trim, equal_var=False)
          assert_allclose(statistic, tr, atol=1e-15)
          assert_allclose(pvalue, pr, atol=1e-15)
  
      def test_compare_SAS(self):
          # Source of the data used in this test:
          # https://support.sas.com/resources/papers/proceedings14/1660-2014.pdf
          a = [12, 14, 18, 25, 32, 44, 12, 14, 18, 25, 32, 44]
          b = [17, 22, 14, 12, 30, 29, 19, 17, 22, 14, 12, 30, 29, 19]
          # In this paper, a trimming percentage of 5% is used. However,
          # in their implementation, the number of values trimmed is rounded to
          # the nearest whole number. However, consistent with
          # `scipy.stats.trimmed_mean`, this test truncates to the lower
          # whole number. In this example, the paper notes that 1 value is
          # trimmed off of each side. 9% replicates this amount of trimming.
          statistic, pvalue = stats.ttest_ind(a, b, trim=.09, equal_var=False)
          assert_allclose(pvalue, 0.514522, atol=1e-6)
          assert_allclose(statistic, 0.669169, atol=1e-6)
  
      def test_equal_var(self):
          '''
          The PairedData library only supports unequal variances. To compare
          samples with equal variances, the multicon library is used.
          > library(multicon)
          > a <- c(2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9)
          > b <- c(6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1)
          > dv = c(a,b)
          > iv = c(rep('a', length(a)), rep('b', length(b)))
          > yuenContrast(dv~ iv, EQVAR = TRUE)
          $Ms
             N                 M wgt
          a 11 2.442857142857143   1
          b 11 5.385714285714286  -1
  
          $test
                                stat df              crit                   p
          results -4.246116897032513 12 2.178812829667228 0.00113508833897713
          '''
          a = [2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9]
          b = [6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1]
          # `equal_var=True` is default
          statistic, pvalue = stats.ttest_ind(a, b, trim=.2)
          assert_allclose(pvalue, 0.00113508833897713, atol=1e-10)
          assert_allclose(statistic, -4.246116897032513, atol=1e-10)
  
      @pytest.mark.parametrize('alt,pr,tr',
                               (('greater', 0.9985605452443, -4.2461168970325),
                                ('less', 0.001439454755672, -4.2461168970325),),
                               )
      def test_alternatives(self, alt, pr, tr):
          '''
          > library(PairedData)
          > a <- c(2.7,2.7,1.1,3.0,1.9,3.0,3.8,3.8,0.3,1.9,1.9)
          > b <- c(6.5,5.4,8.1,3.5,0.5,3.8,6.8,4.9,9.5,6.2,4.1)
          > options(digits=16)
          > yuen.t.test(a, b, alternative = 'greater')
          '''
          a = [2.7, 2.7, 1.1, 3.0, 1.9, 3.0, 3.8, 3.8, 0.3, 1.9, 1.9]
          b = [6.5, 5.4, 8.1, 3.5, 0.5, 3.8, 6.8, 4.9, 9.5, 6.2, 4.1]
  
          statistic, pvalue = stats.ttest_ind(a, b, trim=.2, equal_var=False,
                                              alternative=alt)
          assert_allclose(pvalue, pr, atol=1e-10)
          assert_allclose(statistic, tr, atol=1e-10)
  
      def test_errors_unsupported(self):
          # confirm that attempting to trim with NaNs or permutations raises an
          # error
          match = "Permutations are currently not supported with trimming."
          with assert_raises(ValueError, match=match):
              stats.ttest_ind([1, 2], [2, 3], trim=.2, permutations=2)
          match = ("not supported by permutation tests or trimmed tests.")
          with assert_raises(ValueError, match=match):
              stats.ttest_ind([1, 2], [2, np.nan, 3], trim=.2, nan_policy='omit')
  
      @pytest.mark.parametrize("trim", [-.2, .5, 1])
      def test_trim_bounds_error(self, trim):
          match = "Trimming percentage should be 0 <= `trim` < .5."
          with assert_raises(ValueError, match=match):
              stats.ttest_ind([1, 2], [2, 1], trim=trim)
  
  
  def test__broadcast_concatenate():
      # test that _broadcast_concatenate properly broadcasts arrays along all
      # axes except `axis`, then concatenates along axis
      np.random.seed(0)
      a = np.random.rand(5, 4, 4, 3, 1, 6)
      b = np.random.rand(4, 1, 8, 2, 6)
      c = _broadcast_concatenate((a, b), axis=-3)
      # broadcast manually as an independent check
      a = np.tile(a, (1, 1, 1, 1, 2, 1))
      b = np.tile(b[None, ...], (5, 1, 4, 1, 1, 1))
      for index in product(*(range(i) for i in c.shape)):
          i, j, k, l, m, n = index
          if l < a.shape[-3]:
              assert a[i, j, k, l, m, n] == c[i, j, k, l, m, n]
          else:
              assert b[i, j, k, l - a.shape[-3], m, n] == c[i, j, k, l, m, n]
  
  
  def test_ttest_ind_with_uneq_var():
      # check vs. R
      a = (1, 2, 3)
      b = (1.1, 2.9, 4.2)
      pr = 0.53619490753126731
      tr = -0.68649512735572582
      t, p = stats.ttest_ind(a, b, equal_var=False)
      assert_array_almost_equal([t,p], [tr, pr])
      # test from desc stats API
      assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(a, b),
                                                           equal_var=False),
                                [t, p])
  
      a = (1, 2, 3, 4)
      pr = 0.84354139131608286
      tr = -0.2108663315950719
      t, p = stats.ttest_ind(a, b, equal_var=False)
      assert_array_almost_equal([t,p], [tr, pr])
      assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(a, b),
                                                           equal_var=False),
                                [t, p])
  
      # regression test
      tr = 1.0912746897927283
      tr_uneq_n = 0.66745638708050492
      pr = 0.27647831993021388
      pr_uneq_n = 0.50873585065616544
      tpr = ([tr,-tr],[pr,pr])
  
      rvs3 = np.linspace(1,100, 25)
      rvs2 = np.linspace(1,100,100)
      rvs1 = np.linspace(5,105,100)
      rvs1_2D = np.array([rvs1, rvs2])
  
      rvs2_2D = np.array([rvs2, rvs1])
  
      t,p = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
      assert_array_almost_equal([t,p],(tr,pr))
      assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
                                                                        rvs2),
                                                           equal_var=False),
                                (t, p))
  
      t,p = stats.ttest_ind(rvs1, rvs3, axis=0, equal_var=False)
      assert_array_almost_equal([t,p], (tr_uneq_n, pr_uneq_n))
      assert_array_almost_equal(stats.ttest_ind_from_stats(*_desc_stats(rvs1,
                                                                        rvs3),
                                                           equal_var=False),
                                (t, p))
  
      t,p = stats.ttest_ind(rvs1_2D.T, rvs2_2D.T, axis=0, equal_var=False)
      assert_array_almost_equal([t,p],tpr)
      args = _desc_stats(rvs1_2D.T, rvs2_2D.T)
      assert_array_almost_equal(stats.ttest_ind_from_stats(*args,
                                                           equal_var=False),
                                (t, p))
  
      t,p = stats.ttest_ind(rvs1_2D, rvs2_2D, axis=1, equal_var=False)
      assert_array_almost_equal([t,p],tpr)
      args = _desc_stats(rvs1_2D, rvs2_2D, axis=1)
      assert_array_almost_equal(stats.ttest_ind_from_stats(*args,
                                                           equal_var=False),
                                (t, p))
  
      # test for namedtuple attribute results
      attributes = ('statistic', 'pvalue')
      res = stats.ttest_ind(rvs1, rvs2, axis=0, equal_var=False)
      check_named_results(res, attributes)
  
      # test on 3 dimensions
      rvs1_3D = np.dstack([rvs1_2D,rvs1_2D,rvs1_2D])
      rvs2_3D = np.dstack([rvs2_2D,rvs2_2D,rvs2_2D])
      t,p = stats.ttest_ind(rvs1_3D, rvs2_3D, axis=1, equal_var=False)
      assert_almost_equal(np.abs(t), np.abs(tr))
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (2, 3))
      args = _desc_stats(rvs1_3D, rvs2_3D, axis=1)
      t, p = stats.ttest_ind_from_stats(*args, equal_var=False)
      assert_almost_equal(np.abs(t), np.abs(tr))
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (2, 3))
  
      t, p = stats.ttest_ind(np.moveaxis(rvs1_3D, 2, 0),
                             np.moveaxis(rvs2_3D, 2, 0),
                             axis=2, equal_var=False)
      assert_array_almost_equal(np.abs(t), np.abs(tr))
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (3, 2))
      args = _desc_stats(np.moveaxis(rvs1_3D, 2, 0),
                         np.moveaxis(rvs2_3D, 2, 0), axis=2)
      t, p = stats.ttest_ind_from_stats(*args, equal_var=False)
      assert_array_almost_equal(np.abs(t), np.abs(tr))
      assert_array_almost_equal(np.abs(p), pr)
      assert_equal(t.shape, (3, 2))
  
      # test zero division problem
      with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
          t, p = stats.ttest_ind([0, 0, 0], [1, 1, 1], equal_var=False)
      assert_equal((np.abs(t), p), (np.inf, 0))
      with np.errstate(all='ignore'):
          assert_equal(stats.ttest_ind([0, 0, 0], [0, 0, 0], equal_var=False),
                       (np.nan, np.nan))
  
          # check that nan in input array result in nan output
          anan = np.array([[1, np.nan], [-1, 1]])
          assert_equal(stats.ttest_ind(anan, np.zeros((2, 2)), equal_var=False),
                       ([0, np.nan], [1, np.nan]))
  
  
  def test_ttest_ind_nan_2nd_arg():
      # regression test for gh-6134: nans in the second arg were not handled
      x = [np.nan, 2.0, 3.0, 4.0]
      y = [1.0, 2.0, 1.0, 2.0]
  
      r1 = stats.ttest_ind(x, y, nan_policy='omit')
      r2 = stats.ttest_ind(y, x, nan_policy='omit')
      assert_allclose(r2.statistic, -r1.statistic, atol=1e-15)
      assert_allclose(r2.pvalue, r1.pvalue, atol=1e-15)
  
      # NB: arguments are not paired when NaNs are dropped
      r3 = stats.ttest_ind(y, x[1:])
      assert_allclose(r2, r3, atol=1e-15)
  
      # .. and this is consistent with R. R code:
      # x = c(NA, 2.0, 3.0, 4.0)
      # y = c(1.0, 2.0, 1.0, 2.0)
      # t.test(x, y, var.equal=TRUE)
      assert_allclose(r2, (-2.5354627641855498, 0.052181400457057901),
                      atol=1e-15)
  
  
  def test_ttest_ind_empty_1d_returns_nan():
      # Two empty inputs should return a Ttest_indResult containing nan
      # for both values.
      result = stats.ttest_ind([], [])
      assert isinstance(result, stats._stats_py.Ttest_indResult)
      assert_equal(result, (np.nan, np.nan))
  
  
  @pytest.mark.parametrize('b, expected_shape',
                           [(np.empty((1, 5, 0)), (3, 5)),
                            (np.empty((1, 0, 0)), (3, 0))])
  def test_ttest_ind_axis_size_zero(b, expected_shape):
      # In this test, the length of the axis dimension is zero.
      # The results should be arrays containing nan with shape
      # given by the broadcast nonaxis dimensions.
      a = np.empty((3, 1, 0))
      result = stats.ttest_ind(a, b, axis=-1)
      assert isinstance(result, stats._stats_py.Ttest_indResult)
      expected_value = np.full(expected_shape, fill_value=np.nan)
      assert_equal(result.statistic, expected_value)
      assert_equal(result.pvalue, expected_value)
  
  
  def test_ttest_ind_nonaxis_size_zero():
      # In this test, the length of the axis dimension is nonzero,
      # but one of the nonaxis dimensions has length 0.  Check that
      # we still get the correctly broadcast shape, which is (5, 0)
      # in this case.
      a = np.empty((1, 8, 0))
      b = np.empty((5, 8, 1))
      result = stats.ttest_ind(a, b, axis=1)
      assert isinstance(result, stats._stats_py.Ttest_indResult)
      assert_equal(result.statistic.shape, (5, 0))
      assert_equal(result.pvalue.shape, (5, 0))
  
  
  def test_ttest_ind_nonaxis_size_zero_different_lengths():
      # In this test, the length of the axis dimension is nonzero,
      # and that size is different in the two inputs,
      # and one of the nonaxis dimensions has length 0.  Check that
      # we still get the correctly broadcast shape, which is (5, 0)
      # in this case.
      a = np.empty((1, 7, 0))
      b = np.empty((5, 8, 1))
      result = stats.ttest_ind(a, b, axis=1)
      assert isinstance(result, stats._stats_py.Ttest_indResult)
      assert_equal(result.statistic.shape, (5, 0))
      assert_equal(result.pvalue.shape, (5, 0))
  
  
  def test_gh5686():
      mean1, mean2 = np.array([1, 2]), np.array([3, 4])
      std1, std2 = np.array([5, 3]), np.array([4, 5])
      nobs1, nobs2 = np.array([130, 140]), np.array([100, 150])
      # This will raise a TypeError unless gh-5686 is fixed.
      stats.ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2)
  
  
  def test_ttest_ind_from_stats_inputs_zero():
      # Regression test for gh-6409.
      result = stats.ttest_ind_from_stats(0, 0, 6, 0, 0, 6, equal_var=False)
      assert_equal(result, [np.nan, np.nan])
  
  
  def test_ttest_1samp_new():
      n1, n2, n3 = (10,15,20)
      rvn1 = stats.norm.rvs(loc=5,scale=10,size=(n1,n2,n3))
  
      # check multidimensional array and correct axis handling
      # deterministic rvn1 and rvn2 would be better as in test_ttest_rel
      t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n2,n3)),axis=0)
      t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=0)
      t3,p3 = stats.ttest_1samp(rvn1[:,0,0], 1)
      assert_array_almost_equal(t1,t2, decimal=14)
      assert_almost_equal(t1[0,0],t3, decimal=14)
      assert_equal(t1.shape, (n2,n3))
  
      t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n1,n3)),axis=1)
      t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=1)
      t3,p3 = stats.ttest_1samp(rvn1[0,:,0], 1)
      assert_array_almost_equal(t1,t2, decimal=14)
      assert_almost_equal(t1[0,0],t3, decimal=14)
      assert_equal(t1.shape, (n1,n3))
  
      t1,p1 = stats.ttest_1samp(rvn1[:,:,:], np.ones((n1,n2)),axis=2)
      t2,p2 = stats.ttest_1samp(rvn1[:,:,:], 1,axis=2)
      t3,p3 = stats.ttest_1samp(rvn1[0,0,:], 1)
      assert_array_almost_equal(t1,t2, decimal=14)
      assert_almost_equal(t1[0,0],t3, decimal=14)
      assert_equal(t1.shape, (n1,n2))
  
      # test zero division problem
      t, p = stats.ttest_1samp([0, 0, 0], 1)
      assert_equal((np.abs(t), p), (np.inf, 0))
  
      # test alternative parameter
      # Convert from two-sided p-values to one sided using T result data.
      def convert(t, p, alt):
          if (t < 0 and alt == "less") or (t > 0 and alt == "greater"):
              return p / 2
          return 1 - (p / 2)
      converter = np.vectorize(convert)
      tr, pr = stats.ttest_1samp(rvn1[:, :, :], 1)
  
      t, p = stats.ttest_1samp(rvn1[:, :, :], 1, alternative="greater")
      pc = converter(tr, pr, "greater")
      assert_allclose(p, pc)
      assert_allclose(t, tr)
  
      t, p = stats.ttest_1samp(rvn1[:, :, :], 1, alternative="less")
      pc = converter(tr, pr, "less")
      assert_allclose(p, pc)
      assert_allclose(t, tr)
  
      with np.errstate(all='ignore'):
          assert_equal(stats.ttest_1samp([0, 0, 0], 0), (np.nan, np.nan))
  
          # check that nan in input array result in nan output
          anan = np.array([[1, np.nan],[-1, 1]])
          assert_equal(stats.ttest_1samp(anan, 0), ([0, np.nan], [1, np.nan]))
  
      rvn1[0:2, 1:3, 4:8] = np.nan
  
      tr, pr = stats.ttest_1samp(rvn1[:, :, :], 1, nan_policy='omit')
  
      t, p = stats.ttest_1samp(rvn1[:, :, :], 1, nan_policy='omit',
                               alternative="greater")
      pc = converter(tr, pr, "greater")
      assert_allclose(p, pc)
      assert_allclose(t, tr)
  
      t, p = stats.ttest_1samp(rvn1[:, :, :], 1, nan_policy='omit',
                               alternative="less")
      pc = converter(tr, pr, "less")
      assert_allclose(p, pc)
      assert_allclose(t, tr)
  
  
  class TestDescribe:
      def test_describe_scalar(self):
          with suppress_warnings() as sup, np.errstate(invalid="ignore"), \
               pytest.warns(RuntimeWarning, match="Precision loss occurred"):
              sup.filter(RuntimeWarning, "Degrees of freedom <= 0 for slice")
              n, mm, m, v, sk, kurt = stats.describe(4.)
          assert_equal(n, 1)
          assert_equal(mm, (4.0, 4.0))
          assert_equal(m, 4.0)
          assert np.isnan(v)
          assert np.isnan(sk)
          assert np.isnan(kurt)
  
      def test_describe_numbers(self):
          x = np.vstack((np.ones((3,4)), np.full((2, 4), 2)))
          nc, mmc = (5, ([1., 1., 1., 1.], [2., 2., 2., 2.]))
          mc = np.array([1.4, 1.4, 1.4, 1.4])
          vc = np.array([0.3, 0.3, 0.3, 0.3])
          skc = [0.40824829046386357] * 4
          kurtc = [-1.833333333333333] * 4
          n, mm, m, v, sk, kurt = stats.describe(x)
          assert_equal(n, nc)
          assert_equal(mm, mmc)
          assert_equal(m, mc)
          assert_equal(v, vc)
          assert_array_almost_equal(sk, skc, decimal=13)
          assert_array_almost_equal(kurt, kurtc, decimal=13)
          n, mm, m, v, sk, kurt = stats.describe(x.T, axis=1)
          assert_equal(n, nc)
          assert_equal(mm, mmc)
          assert_equal(m, mc)
          assert_equal(v, vc)
          assert_array_almost_equal(sk, skc, decimal=13)
          assert_array_almost_equal(kurt, kurtc, decimal=13)
  
          x = np.arange(10.)
          x[9] = np.nan
  
          nc, mmc = (9, (0.0, 8.0))
          mc = 4.0
          vc = 7.5
          skc = 0.0
          kurtc = -1.2300000000000002
          n, mm, m, v, sk, kurt = stats.describe(x, nan_policy='omit')
          assert_equal(n, nc)
          assert_equal(mm, mmc)
          assert_equal(m, mc)
          assert_equal(v, vc)
          assert_array_almost_equal(sk, skc)
          assert_array_almost_equal(kurt, kurtc, decimal=13)
  
          assert_raises(ValueError, stats.describe, x, nan_policy='raise')
          assert_raises(ValueError, stats.describe, x, nan_policy='foobar')
  
      def test_describe_result_attributes(self):
          actual = stats.describe(np.arange(5))
          attributes = ('nobs', 'minmax', 'mean', 'variance', 'skewness',
                        'kurtosis')
          check_named_results(actual, attributes)
  
      def test_describe_ddof(self):
          x = np.vstack((np.ones((3, 4)), np.full((2, 4), 2)))
          nc, mmc = (5, ([1., 1., 1., 1.], [2., 2., 2., 2.]))
          mc = np.array([1.4, 1.4, 1.4, 1.4])
          vc = np.array([0.24, 0.24, 0.24, 0.24])
          skc = [0.40824829046386357] * 4
          kurtc = [-1.833333333333333] * 4
          n, mm, m, v, sk, kurt = stats.describe(x, ddof=0)
          assert_equal(n, nc)
          assert_allclose(mm, mmc, rtol=1e-15)
          assert_allclose(m, mc, rtol=1e-15)
          assert_allclose(v, vc, rtol=1e-15)
          assert_array_almost_equal(sk, skc, decimal=13)
          assert_array_almost_equal(kurt, kurtc, decimal=13)
  
      def test_describe_axis_none(self):
          x = np.vstack((np.ones((3, 4)), np.full((2, 4), 2)))
  
          # expected values
          e_nobs, e_minmax = (20, (1.0, 2.0))
          e_mean = 1.3999999999999999
          e_var = 0.25263157894736848
          e_skew = 0.4082482904638634
          e_kurt = -1.8333333333333333
  
          # actual values
          a = stats.describe(x, axis=None)
  
          assert_equal(a.nobs, e_nobs)
          assert_almost_equal(a.minmax, e_minmax)
          assert_almost_equal(a.mean, e_mean)
          assert_almost_equal(a.variance, e_var)
          assert_array_almost_equal(a.skewness, e_skew, decimal=13)
          assert_array_almost_equal(a.kurtosis, e_kurt, decimal=13)
  
      def test_describe_empty(self):
          assert_raises(ValueError, stats.describe, [])
  
  
  def test_normalitytests():
      with pytest.warns(RuntimeWarning, match="Precision loss occurred"):
          assert_raises(ValueError, stats.skewtest, 4.)
          assert_raises(ValueError, stats.kurtosistest, 4.)
          assert_raises(ValueError, stats.normaltest, 4.)
  
      # numbers verified with R: dagoTest in package fBasics
      st_normal, st_skew, st_kurt = (3.92371918, 1.98078826, -0.01403734)
      pv_normal, pv_skew, pv_kurt = (0.14059673, 0.04761502, 0.98880019)
      pv_skew_less, pv_kurt_less = 1 - pv_skew / 2, pv_kurt / 2
      pv_skew_greater, pv_kurt_greater = pv_skew / 2, 1 - pv_kurt / 2
      x = np.array((-2, -1, 0, 1, 2, 3)*4)**2
      attributes = ('statistic', 'pvalue')
  
      assert_array_almost_equal(stats.normaltest(x), (st_normal, pv_normal))
      check_named_results(stats.normaltest(x), attributes)
      assert_array_almost_equal(stats.skewtest(x), (st_skew, pv_skew))
      assert_array_almost_equal(stats.skewtest(x, alternative='less'),
                                (st_skew, pv_skew_less))
      assert_array_almost_equal(stats.skewtest(x, alternative='greater'),
                                (st_skew, pv_skew_greater))
      check_named_results(stats.skewtest(x), attributes)
      assert_array_almost_equal(stats.kurtosistest(x), (st_kurt, pv_kurt))
      assert_array_almost_equal(stats.kurtosistest(x, alternative='less'),
                                (st_kurt, pv_kurt_less))
      assert_array_almost_equal(stats.kurtosistest(x, alternative='greater'),
                                (st_kurt, pv_kurt_greater))
      check_named_results(stats.kurtosistest(x), attributes)
  
      # some more intuitive tests for kurtosistest and skewtest.
      # see gh-13549.
      # skew parameter is 1 > 0
      a1 = stats.skewnorm.rvs(a=1, size=10000, random_state=123)
      pval = stats.skewtest(a1, alternative='greater').pvalue
      assert_almost_equal(pval, 0.0, decimal=5)
      # excess kurtosis of laplace is 3 > 0
      a2 = stats.laplace.rvs(size=10000, random_state=123)
      pval = stats.kurtosistest(a2, alternative='greater').pvalue
      assert_almost_equal(pval, 0.0)
  
      # Test axis=None (equal to axis=0 for 1-D input)
      assert_array_almost_equal(stats.normaltest(x, axis=None),
             (st_normal, pv_normal))
      assert_array_almost_equal(stats.skewtest(x, axis=None),
             (st_skew, pv_skew))
      assert_array_almost_equal(stats.kurtosistest(x, axis=None),
             (st_kurt, pv_kurt))
  
      x = np.arange(10.)
      x[9] = np.nan
      with np.errstate(invalid="ignore"):
          assert_array_equal(stats.skewtest(x), (np.nan, np.nan))
  
      expected = (1.0184643553962129, 0.30845733195153502)
      assert_array_almost_equal(stats.skewtest(x, nan_policy='omit'), expected)
  
      # test alternative with nan_policy='omit'
      a1[10:100] = np.nan
      z, p = stats.skewtest(a1, nan_policy='omit')
      zl, pl = stats.skewtest(a1, nan_policy='omit', alternative='less')
      zg, pg = stats.skewtest(a1, nan_policy='omit', alternative='greater')
      assert_allclose(zl, z, atol=1e-15)
      assert_allclose(zg, z, atol=1e-15)
      assert_allclose(pl, 1 - p/2, atol=1e-15)
      assert_allclose(pg, p/2, atol=1e-15)
  
      with np.errstate(all='ignore'):
          assert_raises(ValueError, stats.skewtest, x, nan_policy='raise')
      assert_raises(ValueError, stats.skewtest, x, nan_policy='foobar')
      assert_raises(ValueError, stats.skewtest, list(range(8)),
                    alternative='foobar')
  
      x = np.arange(30.)
      x[29] = np.nan
      with np.errstate(all='ignore'):
          assert_array_equal(stats.kurtosistest(x), (np.nan, np.nan))
  
      expected = (-2.2683547379505273, 0.023307594135872967)
      assert_array_almost_equal(stats.kurtosistest(x, nan_policy='omit'),
                                expected)
  
      # test alternative with nan_policy='omit'
      a2[10:20] = np.nan
      z, p = stats.kurtosistest(a2[:100], nan_policy='omit')
      zl, pl = stats.kurtosistest(a2[:100], nan_policy='omit',
                                  alternative='less')
      zg, pg = stats.kurtosistest(a2[:100], nan_policy='omit',
                                  alternative='greater')
      assert_allclose(zl, z, atol=1e-15)
      assert_allclose(zg, z, atol=1e-15)
      assert_allclose(pl, 1 - p/2, atol=1e-15)
      assert_allclose(pg, p/2, atol=1e-15)
  
      assert_raises(ValueError, stats.kurtosistest, x, nan_policy='raise')
      assert_raises(ValueError, stats.kurtosistest, x, nan_policy='foobar')
      assert_raises(ValueError, stats.kurtosistest, list(range(20)),
                    alternative='foobar')
  
      with np.errstate(all='ignore'):
          assert_array_equal(stats.normaltest(x), (np.nan, np.nan))
  
      expected = (6.2260409514287449, 0.04446644248650191)
      assert_array_almost_equal(stats.normaltest(x, nan_policy='omit'), expected)
  
      assert_raises(ValueError, stats.normaltest, x, nan_policy='raise')
      assert_raises(ValueError, stats.normaltest, x, nan_policy='foobar')
  
      # regression test for issue gh-9033: x cleary non-normal but power of
      # negtative denom needs to be handled correctly to reject normality
      counts = [128, 0, 58, 7, 0, 41, 16, 0, 0, 167]
      x = np.hstack([np.full(c, i) for i, c in enumerate(counts)])
      assert_equal(stats.kurtosistest(x)[1] < 0.01, True)
  
  
  class TestRankSums:
  
      np.random.seed(0)
      x, y = np.random.rand(2, 10)
  
      @pytest.mark.parametrize('alternative', ['less', 'greater', 'two-sided'])
      def test_ranksums_result_attributes(self, alternative):
          # ranksums pval = mannwhitneyu pval w/out continuity or tie correction
          res1 = stats.ranksums(self.x, self.y,
                                alternative=alternative).pvalue
          res2 = stats.mannwhitneyu(self.x, self.y, use_continuity=False,
                                    alternative=alternative).pvalue
          assert_allclose(res1, res2)
  
      def test_ranksums_named_results(self):
          res = stats.ranksums(self.x, self.y)
          check_named_results(res, ('statistic', 'pvalue'))
  
      def test_input_validation(self):
          with assert_raises(ValueError, match="alternative must be 'less'"):
              stats.ranksums(self.x, self.y, alternative='foobar')
  
  
  class TestJarqueBera:
      def test_jarque_bera_stats(self):
          np.random.seed(987654321)
          x = np.random.normal(0, 1, 100000)
          y = np.random.chisquare(10000, 100000)
          z = np.random.rayleigh(1, 100000)
  
          assert_equal(stats.jarque_bera(x)[0], stats.jarque_bera(x).statistic)
          assert_equal(stats.jarque_bera(x)[1], stats.jarque_bera(x).pvalue)
  
          assert_equal(stats.jarque_bera(y)[0], stats.jarque_bera(y).statistic)
          assert_equal(stats.jarque_bera(y)[1], stats.jarque_bera(y).pvalue)
  
          assert_equal(stats.jarque_bera(z)[0], stats.jarque_bera(z).statistic)
          assert_equal(stats.jarque_bera(z)[1], stats.jarque_bera(z).pvalue)
  
          assert_(stats.jarque_bera(x)[1] > stats.jarque_bera(y)[1])
          assert_(stats.jarque_bera(x).pvalue > stats.jarque_bera(y).pvalue)
  
          assert_(stats.jarque_bera(x)[1] > stats.jarque_bera(z)[1])
          assert_(stats.jarque_bera(x).pvalue > stats.jarque_bera(z).pvalue)
  
          assert_(stats.jarque_bera(y)[1] > stats.jarque_bera(z)[1])
          assert_(stats.jarque_bera(y).pvalue > stats.jarque_bera(z).pvalue)
  
      def test_jarque_bera_array_like(self):
          np.random.seed(987654321)
          x = np.random.normal(0, 1, 100000)
  
          jb_test1 = JB1, p1 = stats.jarque_bera(list(x))
          jb_test2 = JB2, p2 = stats.jarque_bera(tuple(x))
          jb_test3 = JB3, p3 = stats.jarque_bera(x.reshape(2, 50000))
  
          assert_(JB1 == JB2 == JB3 == jb_test1.statistic == jb_test2.statistic == jb_test3.statistic)
          assert_(p1 == p2 == p3 == jb_test1.pvalue == jb_test2.pvalue == jb_test3.pvalue)
  
      def test_jarque_bera_size(self):
          assert_raises(ValueError, stats.jarque_bera, [])
  
  
  def test_skewtest_too_few_samples():
      # Regression test for ticket #1492.
      # skewtest requires at least 8 samples; 7 should raise a ValueError.
      x = np.arange(7.0)
      assert_raises(ValueError, stats.skewtest, x)
  
  
  def test_kurtosistest_too_few_samples():
      # Regression test for ticket #1425.
      # kurtosistest requires at least 5 samples; 4 should raise a ValueError.
      x = np.arange(4.0)
      assert_raises(ValueError, stats.kurtosistest, x)
  
  
  class TestMannWhitneyU:
      X = [19.8958398126694, 19.5452691647182, 19.0577309166425, 21.716543054589,
           20.3269502208702, 20.0009273294025, 19.3440043632957, 20.4216806548105,
           19.0649894736528, 18.7808043120398, 19.3680942943298, 19.4848044069953,
           20.7514611265663, 19.0894948874598, 19.4975522356628, 18.9971170734274,
           20.3239606288208, 20.6921298083835, 19.0724259532507, 18.9825187935021,
           19.5144462609601, 19.8256857844223, 20.5174677102032, 21.1122407995892,
           17.9490854922535, 18.2847521114727, 20.1072217648826, 18.6439891962179,
           20.4970638083542, 19.5567594734914]
  
      Y = [19.2790668029091, 16.993808441865, 18.5416338448258, 17.2634018833575,
           19.1577183624616, 18.5119655377495, 18.6068455037221, 18.8358343362655,
           19.0366413269742, 18.1135025515417, 19.2201873866958, 17.8344909022841,
           18.2894380745856, 18.6661374133922, 19.9688601693252, 16.0672254617636,
           19.00596360572, 19.201561539032, 19.0487501090183, 19.0847908674356]
  
      significant = 14
  
      def test_mannwhitneyu_one_sided(self):
          u1, p1 = stats.mannwhitneyu(self.X, self.Y, alternative='less')
          u2, p2 = stats.mannwhitneyu(self.Y, self.X, alternative='greater')
          u3, p3 = stats.mannwhitneyu(self.X, self.Y, alternative='greater')
          u4, p4 = stats.mannwhitneyu(self.Y, self.X, alternative='less')
  
          assert_equal(p1, p2)
          assert_equal(p3, p4)
          assert_(p1 != p3)
          assert_equal(u1, 498)
          assert_equal(u2, 102)
          assert_equal(u3, 498)
          assert_equal(u4, 102)
          assert_approx_equal(p1, 0.999957683256589, significant=self.significant)
          assert_approx_equal(p3, 4.5941632666275e-05, significant=self.significant)
  
      def test_mannwhitneyu_two_sided(self):
          u1, p1 = stats.mannwhitneyu(self.X, self.Y, alternative='two-sided')
          u2, p2 = stats.mannwhitneyu(self.Y, self.X, alternative='two-sided')
  
          assert_equal(p1, p2)
          assert_equal(u1, 498)
          assert_equal(u2, 102)
          assert_approx_equal(p1, 9.188326533255e-05,
                              significant=self.significant)
  
      def test_mannwhitneyu_no_correct_one_sided(self):
          u1, p1 = stats.mannwhitneyu(self.X, self.Y, False,
                                      alternative='less')
          u2, p2 = stats.mannwhitneyu(self.Y, self.X, False,
                                      alternative='greater')
          u3, p3 = stats.mannwhitneyu(self.X, self.Y, False,
                                      alternative='greater')
          u4, p4 = stats.mannwhitneyu(self.Y, self.X, False,
                                      alternative='less')
  
          assert_equal(p1, p2)
          assert_equal(p3, p4)
          assert_(p1 != p3)
          assert_equal(u1, 498)
          assert_equal(u2, 102)
          assert_equal(u3, 498)
          assert_equal(u4, 102)
          assert_approx_equal(p1, 0.999955905990004, significant=self.significant)
          assert_approx_equal(p3, 4.40940099958089e-05, significant=self.significant)
  
      def test_mannwhitneyu_no_correct_two_sided(self):
          u1, p1 = stats.mannwhitneyu(self.X, self.Y, False,
                                      alternative='two-sided')
          u2, p2 = stats.mannwhitneyu(self.Y, self.X, False,
                                      alternative='two-sided')
  
          assert_equal(p1, p2)
          assert_equal(u1, 498)
          assert_equal(u2, 102)
          assert_approx_equal(p1, 8.81880199916178e-05,
                              significant=self.significant)
  
      def test_mannwhitneyu_ones(self):
          # test for gh-1428
          x = np.array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 2., 1., 1., 2.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 3., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1.])
  
          y = np.array([1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1., 1., 1., 1.,
                        2., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2., 1., 1., 3.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 2., 1., 2., 1.,
                        1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1., 2.,
                        2., 1., 1., 2., 1., 1., 2., 1., 2., 1., 1., 1., 1., 2.,
                        2., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        1., 2., 1., 1., 1., 1., 1., 2., 2., 2., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
                        2., 1., 1., 2., 1., 1., 1., 1., 2., 1., 1., 1., 1., 1.,
                        1., 1., 1., 1., 1., 1., 1., 2., 1., 1., 1., 2., 1., 1.,
                        1., 1., 1., 1.])
  
          # checked against R wilcox.test
          assert_allclose(stats.mannwhitneyu(x, y, alternative='less'),
                          (16980.5, 2.8214327656317373e-005))
          # p-value from R, e.g. wilcox.test(x, y, alternative="g")
          assert_allclose(stats.mannwhitneyu(x, y, alternative='greater'),
                          (16980.5, 0.9999719954296))
          assert_allclose(stats.mannwhitneyu(x, y, alternative='two-sided'),
                          (16980.5, 5.642865531266e-05))
  
      def test_mannwhitneyu_result_attributes(self):
          # test for namedtuple attribute results
          attributes = ('statistic', 'pvalue')
          res = stats.mannwhitneyu(self.X, self.Y, alternative="less")
          check_named_results(res, attributes)
  
  
  def test_pointbiserial():
      # same as mstats test except for the nan
      # Test data: https://web.archive.org/web/20060504220742/https://support.sas.com/ctx/samples/index.jsp?sid=490&tab=output
      x = [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,0,1,0,
           0,0,0,0,1]
      y = [14.8,13.8,12.4,10.1,7.1,6.1,5.8,4.6,4.3,3.5,3.3,3.2,3.0,
           2.8,2.8,2.5,2.4,2.3,2.1,1.7,1.7,1.5,1.3,1.3,1.2,1.2,1.1,
           0.8,0.7,0.6,0.5,0.2,0.2,0.1]
      assert_almost_equal(stats.pointbiserialr(x, y)[0], 0.36149, 5)
  
      # test for namedtuple attribute results
      attributes = ('correlation', 'pvalue')
      res = stats.pointbiserialr(x, y)
      check_named_results(res, attributes)
  
  
  def test_obrientransform():
      # A couple tests calculated by hand.
      x1 = np.array([0, 2, 4])
      t1 = stats.obrientransform(x1)
      expected = [7, -2, 7]
      assert_allclose(t1[0], expected)
  
      x2 = np.array([0, 3, 6, 9])
      t2 = stats.obrientransform(x2)
      expected = np.array([30, 0, 0, 30])
      assert_allclose(t2[0], expected)
  
      # Test two arguments.
      a, b = stats.obrientransform(x1, x2)
      assert_equal(a, t1[0])
      assert_equal(b, t2[0])
  
      # Test three arguments.
      a, b, c = stats.obrientransform(x1, x2, x1)
      assert_equal(a, t1[0])
      assert_equal(b, t2[0])
      assert_equal(c, t1[0])
  
      # This is a regression test to check np.var replacement.
      # The author of this test didn't separately verify the numbers.
      x1 = np.arange(5)
      result = np.array(
        [[5.41666667, 1.04166667, -0.41666667, 1.04166667, 5.41666667],
         [21.66666667, 4.16666667, -1.66666667, 4.16666667, 21.66666667]])
      assert_array_almost_equal(stats.obrientransform(x1, 2*x1), result, decimal=8)
  
      # Example from "O'Brien Test for Homogeneity of Variance"
      # by Herve Abdi.
      values = range(5, 11)
      reps = np.array([5, 11, 9, 3, 2, 2])
      data = np.repeat(values, reps)
      transformed_values = np.array([3.1828, 0.5591, 0.0344,
                                     1.6086, 5.2817, 11.0538])
      expected = np.repeat(transformed_values, reps)
      result = stats.obrientransform(data)
      assert_array_almost_equal(result[0], expected, decimal=4)
  
  
  def check_equal_gmean(array_like, desired, axis=None, dtype=None, rtol=1e-7,
                        weights=None):
      # Note this doesn't test when axis is not specified
      x = stats.gmean(array_like, axis=axis, dtype=dtype, weights=weights)
      assert_allclose(x, desired, rtol=rtol)
      assert_equal(x.dtype, dtype)
  
  
  def check_equal_hmean(array_like, desired, axis=None, dtype=None, rtol=1e-7,
                        weights=None):
      x = stats.hmean(array_like, axis=axis, dtype=dtype, weights=weights)
      assert_allclose(x, desired, rtol=rtol)
      assert_equal(x.dtype, dtype)
  
  
  def check_equal_pmean(array_like, exp, desired, axis=None, dtype=None,
                        rtol=1e-7, weights=None):
      x = stats.pmean(array_like, exp, axis=axis, dtype=dtype, weights=weights)
      assert_allclose(x, desired, rtol=rtol)
      assert_equal(x.dtype, dtype)
  
  
  class TestHarMean:
      def test_1d_list(self):
          #  Test a 1d list
          a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
          desired = 34.1417152147
          check_equal_hmean(a, desired)
  
          a = [1, 2, 3, 4]
          desired = 4. / (1. / 1 + 1. / 2 + 1. / 3 + 1. / 4)
          check_equal_hmean(a, desired)
  
      def test_1d_array(self):
          #  Test a 1d array
          a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
          desired = 34.1417152147
          check_equal_hmean(a, desired)
  
      def test_1d_array_with_zero(self):
          a = np.array([1, 0])
          desired = 0.0
          assert_equal(stats.hmean(a), desired)
  
      def test_1d_array_with_negative_value(self):
          a = np.array([1, 0, -1])
          assert_raises(ValueError, stats.hmean, a)
  
      # Note the next tests use axis=None as default, not axis=0
      def test_2d_list(self):
          #  Test a 2d list
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = 38.6696271841
          check_equal_hmean(a, desired)
  
      def test_2d_array(self):
          #  Test a 2d array
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = 38.6696271841
          check_equal_hmean(np.array(a), desired)
  
      def test_2d_axis0(self):
          #  Test a 2d list with axis=0
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = np.array([22.88135593, 39.13043478, 52.90076336, 65.45454545])
          check_equal_hmean(a, desired, axis=0)
  
      def test_2d_axis0_with_zero(self):
          a = [[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = np.array([22.88135593, 0.0, 52.90076336, 65.45454545])
          assert_allclose(stats.hmean(a, axis=0), desired)
  
      def test_2d_axis1(self):
          #  Test a 2d list with axis=1
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = np.array([19.2, 63.03939962, 103.80078637])
          check_equal_hmean(a, desired, axis=1)
  
      def test_2d_axis1_with_zero(self):
          a = [[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = np.array([0.0, 63.03939962, 103.80078637])
          assert_allclose(stats.hmean(a, axis=1), desired)
  
      def test_weights_1d_list(self):
          # Desired result from:
          # https://www.hackmath.net/en/math-problem/35871
          a = [2, 10, 6]
          weights = [10, 5, 3]
          desired = 3
          check_equal_hmean(a, desired, weights=weights, rtol=1e-5)
  
      def test_weights_2d_array_axis0(self):
          # Desired result from:
          # https://www.hackmath.net/en/math-problem/35871
          a = np.array([[2, 5], [10, 5], [6, 5]])
          weights = np.array([[10, 1], [5, 1], [3, 1]])
          desired = np.array([3, 5])
          check_equal_hmean(a, desired, axis=0, weights=weights, rtol=1e-5)
  
      def test_weights_2d_array_axis1(self):
          # Desired result from:
          # https://www.hackmath.net/en/math-problem/35871
          a = np.array([[2, 10, 6], [7, 7, 7]])
          weights = np.array([[10, 5, 3], [1, 1, 1]])
          desired = np.array([3, 7])
          check_equal_hmean(a, desired, axis=1, weights=weights, rtol=1e-5)
  
      def test_weights_masked_1d_array(self):
          # Desired result from:
          # https://www.hackmath.net/en/math-problem/35871
          a = np.array([2, 10, 6, 42])
          weights = np.ma.array([10, 5, 3, 42], mask=[0, 0, 0, 1])
          desired = 3
          check_equal_hmean(a, desired, weights=weights, rtol=1e-5)
  
  
  class TestGeoMean:
      def test_1d_list(self):
          #  Test a 1d list
          a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]
          desired = 45.2872868812
          check_equal_gmean(a, desired)
  
          a = [1, 2, 3, 4]
          desired = power(1 * 2 * 3 * 4, 1. / 4.)
          check_equal_gmean(a, desired, rtol=1e-14)
  
      def test_1d_array(self):
          #  Test a 1d array
          a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100])
          desired = 45.2872868812
          check_equal_gmean(a, desired)
  
          a = array([1, 2, 3, 4], float32)
          desired = power(1 * 2 * 3 * 4, 1. / 4.)
          check_equal_gmean(a, desired, dtype=float32)
  
      # Note the next tests use axis=None as default, not axis=0
      def test_2d_list(self):
          #  Test a 2d list
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = 52.8885199
          check_equal_gmean(a, desired)
  
      def test_2d_array(self):
          #  Test a 2d array
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = 52.8885199
          check_equal_gmean(array(a), desired)
  
      def test_2d_axis0(self):
          #  Test a 2d list with axis=0
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = np.array([35.56893304, 49.32424149, 61.3579244, 72.68482371])
          check_equal_gmean(a, desired, axis=0)
  
          a = array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]])
          desired = array([1, 2, 3, 4])
          check_equal_gmean(a, desired, axis=0, rtol=1e-14)
  
      def test_2d_axis1(self):
          #  Test a 2d list with axis=1
          a = [[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]]
          desired = np.array([22.13363839, 64.02171746, 104.40086817])
          check_equal_gmean(a, desired, axis=1)
  
          a = array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]])
          v = power(1 * 2 * 3 * 4, 1. / 4.)
          desired = array([v, v, v])
          check_equal_gmean(a, desired, axis=1, rtol=1e-14)
  
      def test_large_values(self):
          a = array([1e100, 1e200, 1e300])
          desired = 1e200
          check_equal_gmean(a, desired, rtol=1e-13)
  
      def test_1d_list0(self):
          #  Test a 1d list with zero element
          a = [10, 20, 30, 40, 50, 60, 70, 80, 90, 0]
          desired = 0.0  # due to exp(-inf)=0
          with np.errstate(all='ignore'):
              check_equal_gmean(a, desired)
  
      def test_1d_array0(self):
          #  Test a 1d array with zero element
          a = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 0])
          desired = 0.0  # due to exp(-inf)=0
          with np.errstate(divide='ignore'):
              check_equal_gmean(a, desired)
  
      def test_1d_list_neg(self):
          #  Test a 1d list with negative element
          a = [10, 20, 30, 40, 50, 60, 70, 80, 90, -1]
          desired = np.nan  # due to log(-1) = nan
          with np.errstate(invalid='ignore'):
              check_equal_gmean(a, desired)
  
      def test_weights_1d_list(self):
          # Desired result from:
          # https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
          a = [1, 2, 3, 4, 5]
          weights = [2, 5, 6, 4, 3]
          desired = 2.77748
          check_equal_gmean(a, desired, weights=weights, rtol=1e-5)
  
      def test_weights_1d_array(self):
          # Desired result from:
          # https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
          a = np.array([1, 2, 3, 4, 5])
          weights = np.array([2, 5, 6, 4, 3])
          desired = 2.77748
          check_equal_gmean(a, desired, weights=weights, rtol=1e-5)
  
      def test_weights_masked_1d_array(self):
          # Desired result from:
          # https://www.dummies.com/education/math/business-statistics/how-to-find-the-weighted-geometric-mean-of-a-data-set/
          a = np.array([1, 2, 3, 4, 5, 6])
          weights = np.ma.array([2, 5, 6, 4, 3, 5], mask=[0, 0, 0, 0, 0, 1])
          desired = 2.77748
          check_equal_gmean(a, desired, weights=weights, rtol=1e-5)
  
  
  class TestPowMean:
  
      def pmean_reference(a, p):
          return (np.sum(a**p) / a.size)**(1/p)
  
      def wpmean_reference(a, p, weights):
          return (np.sum(weights * a**p) / np.sum(weights))**(1/p)
  
      def test_bad_exponent(self):
          with pytest.raises(ValueError, match='Power mean only defined for'):
              stats.pmean([1, 2, 3], [0])
          with pytest.raises(ValueError, match='Power mean only defined for'):
              stats.pmean([1, 2, 3], np.array([0]))
  
      def test_1d_list(self):
          a, p = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], 3.5
          desired = TestPowMean.pmean_reference(np.array(a), p)
          check_equal_pmean(a, p, desired)
  
          a, p = [1, 2, 3, 4], 2
          desired = np.sqrt((1**2 + 2**2 + 3**2 + 4**2) / 4)
          check_equal_pmean(a, p, desired)
  
      def test_1d_array(self):
          a, p = np.array([10, 20, 30, 40, 50, 60, 70, 80, 90, 100]), -2.5
          desired = TestPowMean.pmean_reference(a, p)
          check_equal_pmean(a, p, desired)
  
      def test_1d_array_with_zero(self):
          a, p = np.array([1, 0]), -1
          desired = 0.0
          assert_equal(stats.pmean(a, p), desired)
  
      def test_1d_array_with_negative_value(self):
          a, p = np.array([1, 0, -1]), 1.23
          with pytest.raises(ValueError, match='Power mean only defined if all'):
              stats.pmean(a, p)
  
      @pytest.mark.parametrize(
          ("a", "p"),
          [([[10, 20], [50, 60], [90, 100]], -0.5),
           (np.array([[10, 20], [50, 60], [90, 100]]), 0.5)]
      )
      def test_2d_axisnone(self, a, p):
          desired = TestPowMean.pmean_reference(np.array(a), p)
          check_equal_pmean(a, p, desired)
  
      @pytest.mark.parametrize(
          ("a", "p"),
          [([[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], -0.5),
           ([[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], 0.5)]
      )
      def test_2d_list_axis0(self, a, p):
          desired = [
              TestPowMean.pmean_reference(
                  np.array([a[i][j] for i in range(len(a))]), p
              )
              for j in range(len(a[0]))
          ]
          check_equal_pmean(a, p, desired, axis=0)
  
      @pytest.mark.parametrize(
          ("a", "p"),
          [([[10, 20, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], -0.5),
           ([[10, 0, 30, 40], [50, 60, 70, 80], [90, 100, 110, 120]], 0.5)]
      )
      def test_2d_list_axis1(self, a, p):
          desired = [TestPowMean.pmean_reference(np.array(a_), p) for a_ in a]
          check_equal_pmean(a, p, desired, axis=1)
  
      def test_weights_1d_list(self):
          a, p = [2, 10, 6], -1.23456789
          weights = [10, 5, 3]
          desired = TestPowMean.wpmean_reference(np.array(a), p, weights)
          check_equal_pmean(a, p, desired, weights=weights, rtol=1e-5)
  
      def test_weights_masked_1d_array(self):
          a, p = np.array([2, 10, 6, 42]), 1
          weights = np.ma.array([10, 5, 3, 42], mask=[0, 0, 0, 1])
          desired = np.average(a, weights=weights)
          check_equal_pmean(a, p, desired, weights=weights, rtol=1e-5)
  
      @pytest.mark.parametrize(
          ("axis", "fun_name", "p"),
          [(None, "wpmean_reference", 9.87654321),
           (0, "gmean", 0),
           (1, "hmean", -1)]
      )
      def test_weights_2d_array(self, axis, fun_name, p):
          if fun_name == 'wpmean_reference':
              def fun(a, axis, weights):
                  return TestPowMean.wpmean_reference(a, p, weights)
          else:
              fun = getattr(stats, fun_name)
          a = np.array([[2, 5], [10, 5], [6, 5]])
          weights = np.array([[10, 1], [5, 1], [3, 1]])
          desired = fun(a, axis=axis, weights=weights)
          check_equal_pmean(a, p, desired, axis=axis, weights=weights, rtol=1e-5)
  
  
  class TestGeometricStandardDeviation:
      # must add 1 as `gstd` is only defined for positive values
      array_1d = np.arange(2 * 3 * 4) + 1
      gstd_array_1d = 2.294407613602
      array_3d = array_1d.reshape(2, 3, 4)
  
      def test_1d_array(self):
          gstd_actual = stats.gstd(self.array_1d)
          assert_allclose(gstd_actual, self.gstd_array_1d)
  
      def test_1d_numeric_array_like_input(self):
          gstd_actual = stats.gstd(tuple(self.array_1d))
          assert_allclose(gstd_actual, self.gstd_array_1d)
  
      def test_raises_value_error_non_array_like_input(self):
          with pytest.raises(ValueError, match='Invalid array input'):
              stats.gstd('This should fail as it can not be cast to an array.')
  
      def test_raises_value_error_zero_entry(self):
          with pytest.raises(ValueError, match='Non positive value'):
              stats.gstd(np.append(self.array_1d, [0]))
  
      def test_raises_value_error_negative_entry(self):
          with pytest.raises(ValueError, match='Non positive value'):
              stats.gstd(np.append(self.array_1d, [-1]))
  
      def test_raises_value_error_inf_entry(self):
          with pytest.raises(ValueError, match='Infinite value'):
              stats.gstd(np.append(self.array_1d, [np.inf]))
  
      def test_propagates_nan_values(self):
          a = array([[1, 1, 1, 16], [np.nan, 1, 2, 3]])
          gstd_actual = stats.gstd(a, axis=1)
          assert_allclose(gstd_actual, np.array([4, np.nan]))
  
      def test_ddof_equal_to_number_of_observations(self):
          with pytest.raises(ValueError, match='Degrees of freedom <= 0'):
              stats.gstd(self.array_1d, ddof=self.array_1d.size)
  
      def test_3d_array(self):
          gstd_actual = stats.gstd(self.array_3d, axis=None)
          assert_allclose(gstd_actual, self.gstd_array_1d)
  
      def test_3d_array_axis_type_tuple(self):
          gstd_actual = stats.gstd(self.array_3d, axis=(1,2))
          assert_allclose(gstd_actual, [2.12939215, 1.22120169])
  
      def test_3d_array_axis_0(self):
          gstd_actual = stats.gstd(self.array_3d, axis=0)
          gstd_desired = np.array([
              [6.1330555493918, 3.958900210120, 3.1206598248344, 2.6651441426902],
              [2.3758135028411, 2.174581428192, 2.0260062829505, 1.9115518327308],
              [1.8205343606803, 1.746342404566, 1.6846557065742, 1.6325269194382]
          ])
          assert_allclose(gstd_actual, gstd_desired)
  
      def test_3d_array_axis_1(self):
          gstd_actual = stats.gstd(self.array_3d, axis=1)
          gstd_desired = np.array([
              [3.118993630946, 2.275985934063, 1.933995977619, 1.742896469724],
              [1.271693593916, 1.254158641801, 1.238774141609, 1.225164057869]
          ])
          assert_allclose(gstd_actual, gstd_desired)
  
      def test_3d_array_axis_2(self):
          gstd_actual = stats.gstd(self.array_3d, axis=2)
          gstd_desired = np.array([
              [1.8242475707664, 1.2243686572447, 1.1318311657788],
              [1.0934830582351, 1.0724479791887, 1.0591498540749]
          ])
          assert_allclose(gstd_actual, gstd_desired)
  
      def test_masked_3d_array(self):
          ma = np.ma.masked_where(self.array_3d > 16, self.array_3d)
          gstd_actual = stats.gstd(ma, axis=2)
          gstd_desired = stats.gstd(self.array_3d, axis=2)
          mask = [[0, 0, 0], [0, 1, 1]]
          assert_allclose(gstd_actual, gstd_desired)
          assert_equal(gstd_actual.mask, mask)
  
  
  def test_binomtest():
      # precision tests compared to R for ticket:986
      pp = np.concatenate((np.linspace(0.1, 0.2, 5),
                           np.linspace(0.45, 0.65, 5),
                           np.linspace(0.85, 0.95, 5)))
      n = 501
      x = 450
      results = [0.0, 0.0, 1.0159969301994141e-304,
                 2.9752418572150531e-275, 7.7668382922535275e-250,
                 2.3381250925167094e-099, 7.8284591587323951e-081,
                 9.9155947819961383e-065, 2.8729390725176308e-050,
                 1.7175066298388421e-037, 0.0021070691951093692,
                 0.12044570587262322, 0.88154763174802508, 0.027120993063129286,
                 2.6102587134694721e-006]
  
      for p, res in zip(pp, results):
          assert_approx_equal(stats.binom_test(x, n, p), res,
                              significant=12, err_msg='fail forp=%f' % p)
  
      assert_approx_equal(stats.binom_test(50, 100, 0.1),
                          5.8320387857343647e-024,
                          significant=12)
  
  
  def test_binomtest2():
      # test added for issue #2384
      res2 = [
          [1.0, 1.0],
          [0.5, 1.0, 0.5],
          [0.25, 1.00, 1.00, 0.25],
          [0.125, 0.625, 1.000, 0.625, 0.125],
          [0.0625, 0.3750, 1.0000, 1.0000, 0.3750, 0.0625],
          [0.03125, 0.21875, 0.68750, 1.00000, 0.68750, 0.21875, 0.03125],
          [0.015625, 0.125000, 0.453125, 1.000000, 1.000000, 0.453125, 0.125000,
           0.015625],
          [0.0078125, 0.0703125, 0.2890625, 0.7265625, 1.0000000, 0.7265625,
           0.2890625, 0.0703125, 0.0078125],
          [0.00390625, 0.03906250, 0.17968750, 0.50781250, 1.00000000,
           1.00000000, 0.50781250, 0.17968750, 0.03906250, 0.00390625],
          [0.001953125, 0.021484375, 0.109375000, 0.343750000, 0.753906250,
           1.000000000, 0.753906250, 0.343750000, 0.109375000, 0.021484375,
           0.001953125]
      ]
  
      for k in range(1, 11):
          res1 = [stats.binom_test(v, k, 0.5) for v in range(k + 1)]
          assert_almost_equal(res1, res2[k-1], decimal=10)
  
  
  def test_binomtest3():
      # test added for issue #2384
      # test when x == n*p and neighbors
      res3 = [stats.binom_test(v, v*k, 1./k)
              for v in range(1, 11) for k in range(2, 11)]
      assert_equal(res3, np.ones(len(res3), int))
  
      # > bt=c()
      # > for(i in as.single(1:10)) {
      # +     for(k in as.single(2:10)) {
      # +         bt = c(bt, binom.test(i-1, k*i,(1/k))$p.value);
      # +         print(c(i+1, k*i,(1/k)))
      # +     }
      # + }
      binom_testm1 = np.array([
           0.5, 0.5555555555555556, 0.578125, 0.5904000000000003,
           0.5981224279835393, 0.603430543396034, 0.607304096221924,
           0.610255656871054, 0.612579511000001, 0.625, 0.670781893004115,
           0.68853759765625, 0.6980101120000006, 0.703906431368616,
           0.70793209416498, 0.7108561134173507, 0.713076544331419,
           0.714820192935702, 0.6875, 0.7268709038256367, 0.7418963909149174,
           0.74986110468096, 0.7548015520398076, 0.7581671424768577,
           0.760607984787832, 0.762459425024199, 0.7639120677676575, 0.7265625,
           0.761553963657302, 0.774800934828818, 0.7818005980538996,
           0.78613491480358, 0.789084353140195, 0.7912217659828884,
           0.79284214559524, 0.794112956558801, 0.75390625, 0.7856929451142176,
           0.7976688481430754, 0.8039848974727624, 0.807891868948366,
           0.8105487660137676, 0.812473307174702, 0.8139318233591120,
           0.815075399104785, 0.7744140625, 0.8037322594985427,
           0.814742863657656, 0.8205425178645808, 0.8241275984172285,
           0.8265645374416, 0.8283292196088257, 0.829666291102775,
           0.8307144686362666, 0.7905273437499996, 0.8178712053954738,
           0.828116983756619, 0.833508948940494, 0.8368403871552892,
           0.839104213210105, 0.840743186196171, 0.84198481438049,
           0.8429580531563676, 0.803619384765625, 0.829338573944648,
           0.8389591907548646, 0.84401876783902, 0.84714369697889,
           0.8492667010581667, 0.850803474598719, 0.851967542858308,
           0.8528799045949524, 0.8145294189453126, 0.838881732845347,
           0.847979024541911, 0.852760894015685, 0.8557134656773457,
           0.8577190131799202, 0.85917058278431, 0.860270010472127,
           0.861131648404582, 0.823802947998047, 0.846984756807511,
           0.855635653643743, 0.860180994825685, 0.86298688573253,
           0.864892525675245, 0.866271647085603, 0.867316125625004,
           0.8681346531755114
          ])
  
      # > bt=c()
      # > for(i in as.single(1:10)) {
      # +     for(k in as.single(2:10)) {
      # +         bt = c(bt, binom.test(i+1, k*i,(1/k))$p.value);
      # +         print(c(i+1, k*i,(1/k)))
      # +     }
      # + }
  
      binom_testp1 = np.array([
           0.5, 0.259259259259259, 0.26171875, 0.26272, 0.2632244513031551,
           0.2635138663069203, 0.2636951804161073, 0.2638162407564354,
           0.2639010709000002, 0.625, 0.4074074074074074, 0.42156982421875,
           0.4295746560000003, 0.43473045988554, 0.4383309503172684,
           0.4409884859402103, 0.4430309389962837, 0.444649849401104, 0.6875,
           0.4927602499618962, 0.5096031427383425, 0.5189636628480,
           0.5249280070771274, 0.5290623300865124, 0.5320974248125793,
           0.5344204730474308, 0.536255847400756, 0.7265625, 0.5496019313526808,
           0.5669248746708034, 0.576436455045805, 0.5824538812831795,
           0.5866053321547824, 0.589642781414643, 0.5919618019300193,
           0.593790427805202, 0.75390625, 0.590868349763505, 0.607983393277209,
           0.617303847446822, 0.623172512167948, 0.627208862156123,
           0.6301556891501057, 0.632401894928977, 0.6341708982290303,
           0.7744140625, 0.622562037497196, 0.639236102912278, 0.648263335014579,
           0.65392850011132, 0.657816519817211, 0.660650782947676,
           0.662808780346311, 0.6645068560246006, 0.7905273437499996,
           0.6478843304312477, 0.6640468318879372, 0.6727589686071775,
           0.6782129857784873, 0.681950188903695, 0.684671508668418,
           0.686741824999918, 0.688369886732168, 0.803619384765625,
           0.668716055304315, 0.684360013879534, 0.6927642396829181,
           0.6980155964704895, 0.701609591890657, 0.7042244320992127,
           0.7062125081341817, 0.707775152962577, 0.8145294189453126,
           0.686243374488305, 0.7013873696358975, 0.709501223328243,
           0.714563595144314, 0.718024953392931, 0.7205416252126137,
           0.722454130389843, 0.723956813292035, 0.823802947998047,
           0.701255953767043, 0.715928221686075, 0.723772209289768,
           0.7286603031173616, 0.7319999279787631, 0.7344267920995765,
           0.736270323773157, 0.737718376096348
          ])
  
      res4_p1 = [stats.binom_test(v+1, v*k, 1./k)
                 for v in range(1, 11) for k in range(2, 11)]
      res4_m1 = [stats.binom_test(v-1, v*k, 1./k)
                 for v in range(1, 11) for k in range(2, 11)]
  
      assert_almost_equal(res4_p1, binom_testp1, decimal=13)
      assert_almost_equal(res4_m1, binom_testm1, decimal=13)
  
  
  class TestTrim:
      # test trim functions
      def test_trim1(self):
          a = np.arange(11)
          assert_equal(np.sort(stats.trim1(a, 0.1)), np.arange(10))
          assert_equal(np.sort(stats.trim1(a, 0.2)), np.arange(9))
          assert_equal(np.sort(stats.trim1(a, 0.2, tail='left')),
                       np.arange(2, 11))
          assert_equal(np.sort(stats.trim1(a, 3/11., tail='left')),
                       np.arange(3, 11))
          assert_equal(stats.trim1(a, 1.0), [])
          assert_equal(stats.trim1(a, 1.0, tail='left'), [])
  
          # empty input
          assert_equal(stats.trim1([], 0.1), [])
          assert_equal(stats.trim1([], 3/11., tail='left'), [])
          assert_equal(stats.trim1([], 4/6.), [])
  
          # test axis
          a = np.arange(24).reshape(6, 4)
          ref = np.arange(4, 24).reshape(5, 4)  # first row trimmed
  
          axis = 0
          trimmed = stats.trim1(a, 0.2, tail='left', axis=axis)
          assert_equal(np.sort(trimmed, axis=axis), ref)
  
          axis = 1
          trimmed = stats.trim1(a.T, 0.2, tail='left', axis=axis)
          assert_equal(np.sort(trimmed, axis=axis), ref.T)
  
      def test_trimboth(self):
          a = np.arange(11)
          assert_equal(np.sort(stats.trimboth(a, 3/11.)), np.arange(3, 8))
          assert_equal(np.sort(stats.trimboth(a, 0.2)),
                       np.array([2, 3, 4, 5, 6, 7, 8]))
          assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(6, 4), 0.2)),
                       np.arange(4, 20).reshape(4, 4))
          assert_equal(np.sort(stats.trimboth(np.arange(24).reshape(4, 6).T,
                                              2/6.)),
                       np.array([[2, 8, 14, 20], [3, 9, 15, 21]]))
          assert_raises(ValueError, stats.trimboth,
                        np.arange(24).reshape(4, 6).T, 4/6.)
  
          # empty input
          assert_equal(stats.trimboth([], 0.1), [])
          assert_equal(stats.trimboth([], 3/11.), [])
          assert_equal(stats.trimboth([], 4/6.), [])
  
      def test_trim_mean(self):
          # don't use pre-sorted arrays
          a = np.array([4, 8, 2, 0, 9, 5, 10, 1, 7, 3, 6])
          idx = np.array([3, 5, 0, 1, 2, 4])
          a2 = np.arange(24).reshape(6, 4)[idx, :]
          a3 = np.arange(24).reshape(6, 4, order='F')[idx, :]
          assert_equal(stats.trim_mean(a3, 2/6.),
                       np.array([2.5, 8.5, 14.5, 20.5]))
          assert_equal(stats.trim_mean(a2, 2/6.),
                       np.array([10., 11., 12., 13.]))
          idx4 = np.array([1, 0, 3, 2])
          a4 = np.arange(24).reshape(4, 6)[idx4, :]
          assert_equal(stats.trim_mean(a4, 2/6.),
                       np.array([9., 10., 11., 12., 13., 14.]))
          # shuffled arange(24) as array_like
          a = [7, 11, 12, 21, 16, 6, 22, 1, 5, 0, 18, 10, 17, 9, 19, 15, 23,
               20, 2, 14, 4, 13, 8, 3]
          assert_equal(stats.trim_mean(a, 2/6.), 11.5)
          assert_equal(stats.trim_mean([5,4,3,1,2,0], 2/6.), 2.5)
  
          # check axis argument
          np.random.seed(1234)
          a = np.random.randint(20, size=(5, 6, 4, 7))
          for axis in [0, 1, 2, 3, -1]:
              res1 = stats.trim_mean(a, 2/6., axis=axis)
              res2 = stats.trim_mean(np.moveaxis(a, axis, 0), 2/6.)
              assert_equal(res1, res2)
  
          res1 = stats.trim_mean(a, 2/6., axis=None)
          res2 = stats.trim_mean(a.ravel(), 2/6.)
          assert_equal(res1, res2)
  
          assert_raises(ValueError, stats.trim_mean, a, 0.6)
  
          # empty input
          assert_equal(stats.trim_mean([], 0.0), np.nan)
          assert_equal(stats.trim_mean([], 0.6), np.nan)
  
  
  class TestSigmaClip:
      def test_sigmaclip1(self):
          a = np.concatenate((np.linspace(9.5, 10.5, 31), np.linspace(0, 20, 5)))
          fact = 4  # default
          c, low, upp = stats.sigmaclip(a)
          assert_(c.min() > low)
          assert_(c.max() < upp)
          assert_equal(low, c.mean() - fact*c.std())
          assert_equal(upp, c.mean() + fact*c.std())
          assert_equal(c.size, a.size)
  
      def test_sigmaclip2(self):
          a = np.concatenate((np.linspace(9.5, 10.5, 31), np.linspace(0, 20, 5)))
          fact = 1.5
          c, low, upp = stats.sigmaclip(a, fact, fact)
          assert_(c.min() > low)
          assert_(c.max() < upp)
          assert_equal(low, c.mean() - fact*c.std())
          assert_equal(upp, c.mean() + fact*c.std())
          assert_equal(c.size, 4)
          assert_equal(a.size, 36)  # check original array unchanged
  
      def test_sigmaclip3(self):
          a = np.concatenate((np.linspace(9.5, 10.5, 11),
                              np.linspace(-100, -50, 3)))
          fact = 1.8
          c, low, upp = stats.sigmaclip(a, fact, fact)
          assert_(c.min() > low)
          assert_(c.max() < upp)
          assert_equal(low, c.mean() - fact*c.std())
          assert_equal(upp, c.mean() + fact*c.std())
          assert_equal(c, np.linspace(9.5, 10.5, 11))
  
      def test_sigmaclip_result_attributes(self):
          a = np.concatenate((np.linspace(9.5, 10.5, 11),
                              np.linspace(-100, -50, 3)))
          fact = 1.8
          res = stats.sigmaclip(a, fact, fact)
          attributes = ('clipped', 'lower', 'upper')
          check_named_results(res, attributes)
  
      def test_std_zero(self):
          # regression test #8632
          x = np.ones(10)
          assert_equal(stats.sigmaclip(x)[0], x)
  
  
  class TestAlexanderGovern:
      def test_compare_dtypes(self):
          args = [[13, 13, 13, 13, 13, 13, 13, 12, 12],
                  [14, 13, 12, 12, 12, 12, 12, 11, 11],
                  [14, 14, 13, 13, 13, 13, 13, 12, 12],
                  [15, 14, 13, 13, 13, 12, 12, 12, 11]]
          args_int16 = np.array(args, dtype=np.int16)
          args_int32 = np.array(args, dtype=np.int32)
          args_uint8 = np.array(args, dtype=np.uint8)
          args_float64 = np.array(args, dtype=np.float64)
  
          res_int16 = stats.alexandergovern(*args_int16)
          res_int32 = stats.alexandergovern(*args_int32)
          res_unit8 = stats.alexandergovern(*args_uint8)
          res_float64 = stats.alexandergovern(*args_float64)
  
          assert (res_int16.pvalue == res_int32.pvalue ==
                  res_unit8.pvalue == res_float64.pvalue)
          assert (res_int16.statistic == res_int32.statistic ==
                  res_unit8.statistic == res_float64.statistic)
  
      def test_bad_inputs(self):
          # input array is of size zero
          with assert_raises(ValueError, match="Input sample size must be"
                                               " greater than one."):
              stats.alexandergovern([1, 2], [])
          # input is a singular non list element
          with assert_raises(ValueError, match="Input sample size must be"
                                               " greater than one."):
              stats.alexandergovern([1, 2], 2)
          # input list is of size 1
          with assert_raises(ValueError, match="Input sample size must be"
                                               " greater than one."):
              stats.alexandergovern([1, 2], [2])
          # inputs are not finite (infinity)
          with assert_raises(ValueError, match="Input samples must be finite."):
              stats.alexandergovern([1, 2], [np.inf, np.inf])
          # inputs are multidimensional
          with assert_raises(ValueError, match="Input samples must be one"
                                               "-dimensional"):
              stats.alexandergovern([1, 2], [[1, 2], [3, 4]])
  
      def test_compare_r(self):
          '''
          Data generated in R with
          > set.seed(1)
          > library("onewaytests")
          > library("tibble")
          > y <- c(rnorm(40, sd=10),
          +        rnorm(30, sd=15),
          +        rnorm(20, sd=20))
          > x <- c(rep("one", times=40),
          +        rep("two", times=30),
          +        rep("eight", times=20))
          > x <- factor(x)
          > ag.test(y ~ x, tibble(y,x))
  
          Alexander-Govern Test (alpha = 0.05)
          -------------------------------------------------------------
          data : y and x
  
          statistic  : 1.359941
          parameter  : 2
          p.value    : 0.5066321
  
          Result     : Difference is not statistically significant.
          -------------------------------------------------------------
          Example adapted from:
          https://eval-serv2.metpsy.uni-jena.de/wiki-metheval-hp/index.php/R_FUN_Alexander-Govern
  
          '''
          one = [-6.264538107423324, 1.8364332422208225, -8.356286124100471,
                 15.952808021377916, 3.295077718153605, -8.204683841180152,
                 4.874290524284853, 7.383247051292173, 5.757813516534923,
                 -3.0538838715635603, 15.11781168450848, 3.898432364114311,
                 -6.2124058054180376, -22.146998871774997, 11.249309181431082,
                 -0.4493360901523085, -0.16190263098946087, 9.438362106852992,
                 8.212211950980885, 5.939013212175088, 9.189773716082183,
                 7.821363007310671, 0.745649833651906, -19.89351695863373,
                 6.198257478947102, -0.5612873952900078, -1.557955067053293,
                 -14.707523838992744, -4.781500551086204, 4.179415601997024,
                 13.58679551529044, -1.0278772734299553, 3.876716115593691,
                 -0.5380504058290512, -13.770595568286065, -4.149945632996798,
                 -3.942899537103493, -0.5931339671118566, 11.000253719838831,
                 7.631757484575442]
  
          two = [-2.4678539438038034, -3.8004252020476135, 10.454450631071062,
                 8.34994798010486, -10.331335418242798, -10.612427354431794,
                 5.468729432052455, 11.527993867731237, -1.6851931822534207,
                 13.216615896813222, 5.971588205506021, -9.180395898761569,
                 5.116795371366372, -16.94044644121189, 21.495355525515556,
                 29.7059984775879, -5.508322146997636, -15.662019394747961,
                 8.545794411636193, -2.0258190582123654, 36.024266407571645,
                 -0.5886000409975387, 10.346090436761651, 0.4200323817099909,
                 -11.14909813323608, 2.8318844927151434, -27.074379433365568,
                 21.98332292344329, 2.2988000731784655, 32.58917505543229]
  
          eight = [9.510190577993251, -14.198928618436291, 12.214527069781099,
                   -18.68195263288503, -25.07266800478204, 5.828924710349257,
                   -8.86583746436866, 0.02210703263248262, 1.4868264830332811,
                   -11.79041892376144, -11.37337465637004, -2.7035723024766414,
                   23.56173993146409, -30.47133600859524, 11.878923752568431,
                   6.659007424270365, 21.261996745527256, -6.083678472686013,
                   7.400376198325763, 5.341975815444621]
          soln = stats.alexandergovern(one, two, eight)
          assert_allclose(soln.statistic, 1.3599405447999450836)
          assert_allclose(soln.pvalue, 0.50663205309676440091)
  
      def test_compare_scholar(self):
          '''
          Data taken from 'The Modification and Evaluation of the
          Alexander-Govern Test in Terms of Power' by Kingsley Ochuko, T.,
          Abdullah, S., Binti Zain, Z., & Soaad Syed Yahaya, S. (2015).
          '''
          young = [482.43, 484.36, 488.84, 495.15, 495.24, 502.69, 504.62,
                   518.29, 519.1, 524.1, 524.12, 531.18, 548.42, 572.1, 584.68,
                   609.09, 609.53, 666.63, 676.4]
          middle = [335.59, 338.43, 353.54, 404.27, 437.5, 469.01, 485.85,
                    487.3, 493.08, 494.31, 499.1, 886.41]
          old = [519.01, 528.5, 530.23, 536.03, 538.56, 538.83, 557.24, 558.61,
                 558.95, 565.43, 586.39, 594.69, 629.22, 645.69, 691.84]
          soln = stats.alexandergovern(young, middle, old)
          assert_allclose(soln.statistic, 5.3237, atol=1e-3)
          assert_allclose(soln.pvalue, 0.06982, atol=1e-4)
  
          # verify with ag.test in r
          '''
          > library("onewaytests")
          > library("tibble")
          > young <- c(482.43, 484.36, 488.84, 495.15, 495.24, 502.69, 504.62,
          +                  518.29, 519.1, 524.1, 524.12, 531.18, 548.42, 572.1,
          +                  584.68, 609.09, 609.53, 666.63, 676.4)
          > middle <- c(335.59, 338.43, 353.54, 404.27, 437.5, 469.01, 485.85,
          +                   487.3, 493.08, 494.31, 499.1, 886.41)
          > old <- c(519.01, 528.5, 530.23, 536.03, 538.56, 538.83, 557.24,
          +                   558.61, 558.95, 565.43, 586.39, 594.69, 629.22,
          +                   645.69, 691.84)
          > young_fct <- c(rep("young", times=19))
          > middle_fct <-c(rep("middle", times=12))
          > old_fct <- c(rep("old", times=15))
          > ag.test(a ~ b, tibble(a=c(young, middle, old), b=factor(c(young_fct,
          +                                              middle_fct, old_fct))))
  
          Alexander-Govern Test (alpha = 0.05)
          -------------------------------------------------------------
          data : a and b
  
          statistic  : 5.324629
          parameter  : 2
          p.value    : 0.06978651
  
          Result     : Difference is not statistically significant.
          -------------------------------------------------------------
  
          '''
          assert_allclose(soln.statistic, 5.324629)
          assert_allclose(soln.pvalue, 0.06978651)
  
      def test_compare_scholar3(self):
          '''
          Data taken from 'Robustness And Comparative Power Of WelchAspin,
          Alexander-Govern And Yuen Tests Under Non-Normality And Variance
          Heteroscedasticity', by Ayed A. Almoied. 2017. Page 34-37.
          https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=2775&context=oa_dissertations
          '''
          x1 = [-1.77559, -1.4113, -0.69457, -0.54148, -0.18808, -0.07152,
                0.04696, 0.051183, 0.148695, 0.168052, 0.422561, 0.458555,
                0.616123, 0.709968, 0.839956, 0.857226, 0.929159, 0.981442,
                0.999554, 1.642958]
          x2 = [-1.47973, -1.2722, -0.91914, -0.80916, -0.75977, -0.72253,
                -0.3601, -0.33273, -0.28859, -0.09637, -0.08969, -0.01824,
                0.260131, 0.289278, 0.518254, 0.683003, 0.877618, 1.172475,
                1.33964, 1.576766]
          soln = stats.alexandergovern(x1, x2)
          assert_allclose(soln.statistic, 0.713526, atol=1e-5)
          assert_allclose(soln.pvalue, 0.398276, atol=1e-5)
  
          '''
          tested in ag.test in R:
          > library("onewaytests")
          > library("tibble")
          > x1 <- c(-1.77559, -1.4113, -0.69457, -0.54148, -0.18808, -0.07152,
          +          0.04696, 0.051183, 0.148695, 0.168052, 0.422561, 0.458555,
          +          0.616123, 0.709968, 0.839956, 0.857226, 0.929159, 0.981442,
          +          0.999554, 1.642958)
          > x2 <- c(-1.47973, -1.2722, -0.91914, -0.80916, -0.75977, -0.72253,
          +         -0.3601, -0.33273, -0.28859, -0.09637, -0.08969, -0.01824,
          +         0.260131, 0.289278, 0.518254, 0.683003, 0.877618, 1.172475,
          +         1.33964, 1.576766)
          > x1_fact <- c(rep("x1", times=20))
          > x2_fact <- c(rep("x2", times=20))
          > a <- c(x1, x2)
          > b <- factor(c(x1_fact, x2_fact))
          > ag.test(a ~ b, tibble(a, b))
          Alexander-Govern Test (alpha = 0.05)
          -------------------------------------------------------------
          data : a and b
  
          statistic  : 0.7135182
          parameter  : 1
          p.value    : 0.3982783
  
          Result     : Difference is not statistically significant.
          -------------------------------------------------------------
          '''
          assert_allclose(soln.statistic, 0.7135182)
          assert_allclose(soln.pvalue, 0.3982783)
  
      def test_nan_policy_propogate(self):
          args = [[1, 2, 3, 4], [1, np.nan]]
          # default nan_policy is 'propagate'
          res = stats.alexandergovern(*args)
          assert_equal(res.pvalue, np.nan)
          assert_equal(res.statistic, np.nan)
  
      def test_nan_policy_raise(self):
          args = [[1, 2, 3, 4], [1, np.nan]]
          with assert_raises(ValueError, match="The input contains nan values"):
              stats.alexandergovern(*args, nan_policy='raise')
  
      def test_nan_policy_omit(self):
          args_nan = [[1, 2, 3, None, 4], [1, np.nan, 19, 25]]
          args_no_nan = [[1, 2, 3, 4], [1, 19, 25]]
          res_nan = stats.alexandergovern(*args_nan, nan_policy='omit')
          res_no_nan = stats.alexandergovern(*args_no_nan)
          assert_equal(res_nan.pvalue, res_no_nan.pvalue)
          assert_equal(res_nan.statistic, res_no_nan.statistic)
  
      def test_constant_input(self):
          # Zero variance input, consistent with `stats.pearsonr`
          msg = "An input array is constant; the statistic is not defined."
          with assert_warns(stats.ConstantInputWarning, match=msg):
              res = stats.alexandergovern([0.667, 0.667, 0.667],
                                          [0.123, 0.456, 0.789])
              assert_equal(res.statistic, np.nan)
              assert_equal(res.pvalue, np.nan)
  
  
  class TestFOneWay:
  
      def test_trivial(self):
          # A trivial test of stats.f_oneway, with F=0.
          F, p = stats.f_oneway([0, 2], [0, 2])
          assert_equal(F, 0.0)
          assert_equal(p, 1.0)
  
      def test_basic(self):
          # Despite being a floating point calculation, this data should
          # result in F being exactly 2.0.
          F, p = stats.f_oneway([0, 2], [2, 4])
          assert_equal(F, 2.0)
          assert_allclose(p, 1 - np.sqrt(0.5), rtol=1e-14)
  
      def test_known_exact(self):
          # Another trivial dataset for which the exact F and p can be
          # calculated.
          F, p = stats.f_oneway([2], [2], [2, 3, 4])
          # The use of assert_equal might be too optimistic, but the calculation
          # in this case is trivial enough that it is likely to go through with
          # no loss of precision.
          assert_equal(F, 3/5)
          assert_equal(p, 5/8)
  
      def test_large_integer_array(self):
          a = np.array([655, 788], dtype=np.uint16)
          b = np.array([789, 772], dtype=np.uint16)
          F, p = stats.f_oneway(a, b)
          # The expected value was verified by computing it with mpmath with
          # 40 digits of precision.
          assert_allclose(F, 0.77450216931805540, rtol=1e-14)
  
      def test_result_attributes(self):
          a = np.array([655, 788], dtype=np.uint16)
          b = np.array([789, 772], dtype=np.uint16)
          res = stats.f_oneway(a, b)
          attributes = ('statistic', 'pvalue')
          check_named_results(res, attributes)
  
      def test_nist(self):
          # These are the nist ANOVA files. They can be found at:
          # https://www.itl.nist.gov/div898/strd/anova/anova.html
          filenames = ['SiRstv.dat', 'SmLs01.dat', 'SmLs02.dat', 'SmLs03.dat',
                       'AtmWtAg.dat', 'SmLs04.dat', 'SmLs05.dat', 'SmLs06.dat',
                       'SmLs07.dat', 'SmLs08.dat', 'SmLs09.dat']
  
          for test_case in filenames:
              rtol = 1e-7
              fname = os.path.abspath(os.path.join(os.path.dirname(__file__),
                                                   'data/nist_anova', test_case))
              with open(fname, 'r') as f:
                  content = f.read().split('\n')
              certified = [line.split() for line in content[40:48]
                           if line.strip()]
              dataf = np.loadtxt(fname, skiprows=60)
              y, x = dataf.T
              y = y.astype(int)
              caty = np.unique(y)
              f = float(certified[0][-1])
  
              xlist = [x[y == i] for i in caty]
              res = stats.f_oneway(*xlist)
  
              # With the hard test cases we relax the tolerance a bit.
              hard_tc = ('SmLs07.dat', 'SmLs08.dat', 'SmLs09.dat')
              if test_case in hard_tc:
                  rtol = 1e-4
  
              assert_allclose(res[0], f, rtol=rtol,
                              err_msg='Failing testcase: %s' % test_case)
  
      @pytest.mark.parametrize("a, b, expected", [
          (np.array([42, 42, 42]), np.array([7, 7, 7]), (np.inf, 0)),
          (np.array([42, 42, 42]), np.array([42, 42, 42]), (np.nan, np.nan))
          ])
      def test_constant_input(self, a, b, expected):
          # For more details, look on https://github.com/scipy/scipy/issues/11669
          with assert_warns(stats.F_onewayConstantInputWarning):
              f, p = stats.f_oneway(a, b)
              assert f, p == expected
  
      @pytest.mark.parametrize('axis', [-2, -1, 0, 1])
      def test_2d_inputs(self, axis):
          a = np.array([[1, 4, 3, 3],
                        [2, 5, 3, 3],
                        [3, 6, 3, 3],
                        [2, 3, 3, 3],
                        [1, 4, 3, 3]])
          b = np.array([[3, 1, 5, 3],
                        [4, 6, 5, 3],
                        [4, 3, 5, 3],
                        [1, 5, 5, 3],
                        [5, 5, 5, 3],
                        [2, 3, 5, 3],
                        [8, 2, 5, 3],
                        [2, 2, 5, 3]])
          c = np.array([[4, 3, 4, 3],
                        [4, 2, 4, 3],
                        [5, 4, 4, 3],
                        [5, 4, 4, 3]])
  
          if axis in [-1, 1]:
              a = a.T
              b = b.T
              c = c.T
              take_axis = 0
          else:
              take_axis = 1
  
          with assert_warns(stats.F_onewayConstantInputWarning):
              f, p = stats.f_oneway(a, b, c, axis=axis)
  
          # Verify that the result computed with the 2d arrays matches
          # the result of calling f_oneway individually on each slice.
          for j in [0, 1]:
              fj, pj = stats.f_oneway(np.take(a, j, take_axis),
                                      np.take(b, j, take_axis),
                                      np.take(c, j, take_axis))
              assert_allclose(f[j], fj, rtol=1e-14)
              assert_allclose(p[j], pj, rtol=1e-14)
          for j in [2, 3]:
              with assert_warns(stats.F_onewayConstantInputWarning):
                  fj, pj = stats.f_oneway(np.take(a, j, take_axis),
                                          np.take(b, j, take_axis),
                                          np.take(c, j, take_axis))
                  assert_equal(f[j], fj)
                  assert_equal(p[j], pj)
  
      def test_3d_inputs(self):
          # Some 3-d arrays. (There is nothing special about the values.)
          a = 1/np.arange(1.0, 4*5*7 + 1).reshape(4, 5, 7)
          b = 2/np.arange(1.0, 4*8*7 + 1).reshape(4, 8, 7)
          c = np.cos(1/np.arange(1.0, 4*4*7 + 1).reshape(4, 4, 7))
  
          f, p = stats.f_oneway(a, b, c, axis=1)
  
          assert f.shape == (4, 7)
          assert p.shape == (4, 7)
  
          for i in range(a.shape[0]):
              for j in range(a.shape[2]):
                  fij, pij = stats.f_oneway(a[i, :, j], b[i, :, j], c[i, :, j])
                  assert_allclose(fij, f[i, j])
                  assert_allclose(pij, p[i, j])
  
      def test_length0_1d_error(self):
          # Require at least one value in each group.
          with assert_warns(stats.F_onewayBadInputSizesWarning):
              result = stats.f_oneway([1, 2, 3], [], [4, 5, 6, 7])
              assert_equal(result, (np.nan, np.nan))
  
      def test_length0_2d_error(self):
          with assert_warns(stats.F_onewayBadInputSizesWarning):
              ncols = 3
              a = np.ones((4, ncols))
              b = np.ones((0, ncols))
              c = np.ones((5, ncols))
              f, p = stats.f_oneway(a, b, c)
              nans = np.full((ncols,), fill_value=np.nan)
              assert_equal(f, nans)
              assert_equal(p, nans)
  
      def test_all_length_one(self):
          with assert_warns(stats.F_onewayBadInputSizesWarning):
              result = stats.f_oneway([10], [11], [12], [13])
              assert_equal(result, (np.nan, np.nan))
  
      @pytest.mark.parametrize('args', [(), ([1, 2, 3],)])
      def test_too_few_inputs(self, args):
          with assert_raises(TypeError):
              stats.f_oneway(*args)
  
      def test_axis_error(self):
          a = np.ones((3, 4))
          b = np.ones((5, 4))
          with assert_raises(np.AxisError):
              stats.f_oneway(a, b, axis=2)
  
      def test_bad_shapes(self):
          a = np.ones((3, 4))
          b = np.ones((5, 4))
          with assert_raises(ValueError):
              stats.f_oneway(a, b, axis=1)
  
  
  class TestKruskal:
      def test_simple(self):
          x = [1]
          y = [2]
          h, p = stats.kruskal(x, y)
          assert_equal(h, 1.0)
          assert_approx_equal(p, stats.distributions.chi2.sf(h, 1))
          h, p = stats.kruskal(np.array(x), np.array(y))
          assert_equal(h, 1.0)
          assert_approx_equal(p, stats.distributions.chi2.sf(h, 1))
  
      def test_basic(self):
          x = [1, 3, 5, 7, 9]
          y = [2, 4, 6, 8, 10]
          h, p = stats.kruskal(x, y)
          assert_approx_equal(h, 3./11, significant=10)
          assert_approx_equal(p, stats.distributions.chi2.sf(3./11, 1))
          h, p = stats.kruskal(np.array(x), np.array(y))
          assert_approx_equal(h, 3./11, significant=10)
          assert_approx_equal(p, stats.distributions.chi2.sf(3./11, 1))
  
      def test_simple_tie(self):
          x = [1]
          y = [1, 2]
          h_uncorr = 1.5**2 + 2*2.25**2 - 12
          corr = 0.75
          expected = h_uncorr / corr   # 0.5
          h, p = stats.kruskal(x, y)
          # Since the expression is simple and the exact answer is 0.5, it
          # should be safe to use assert_equal().
          assert_equal(h, expected)
  
      def test_another_tie(self):
          x = [1, 1, 1, 2]
          y = [2, 2, 2, 2]
          h_uncorr = (12. / 8. / 9.) * 4 * (3**2 + 6**2) - 3 * 9
          corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
          expected = h_uncorr / corr
          h, p = stats.kruskal(x, y)
          assert_approx_equal(h, expected)
  
      def test_three_groups(self):
          # A test of stats.kruskal with three groups, with ties.
          x = [1, 1, 1]
          y = [2, 2, 2]
          z = [2, 2]
          h_uncorr = (12. / 8. / 9.) * (3*2**2 + 3*6**2 + 2*6**2) - 3 * 9  # 5.0
          corr = 1 - float(3**3 - 3 + 5**3 - 5) / (8**3 - 8)
          expected = h_uncorr / corr  # 7.0
          h, p = stats.kruskal(x, y, z)
          assert_approx_equal(h, expected)
          assert_approx_equal(p, stats.distributions.chi2.sf(h, 2))
  
      def test_empty(self):
          # A test of stats.kruskal with three groups, with ties.
          x = [1, 1, 1]
          y = [2, 2, 2]
          z = []
          assert_equal(stats.kruskal(x, y, z), (np.nan, np.nan))
  
      def test_kruskal_result_attributes(self):
          x = [1, 3, 5, 7, 9]
          y = [2, 4, 6, 8, 10]
          res = stats.kruskal(x, y)
          attributes = ('statistic', 'pvalue')
          check_named_results(res, attributes)
  
      def test_nan_policy(self):
          x = np.arange(10.)
          x[9] = np.nan
          assert_equal(stats.kruskal(x, x), (np.nan, np.nan))
          assert_almost_equal(stats.kruskal(x, x, nan_policy='omit'), (0.0, 1.0))
          assert_raises(ValueError, stats.kruskal, x, x, nan_policy='raise')
          assert_raises(ValueError, stats.kruskal, x, x, nan_policy='foobar')
  
      def test_large_no_samples(self):
          # Test to see if large samples are handled correctly.
          n = 50000
          x = np.random.randn(n)
          y = np.random.randn(n) + 50
          h, p = stats.kruskal(x, y)
          expected = 0
          assert_approx_equal(p, expected)
  
  
  class TestCombinePvalues:
  
      def test_fisher(self):
          # Example taken from https://en.wikipedia.org/wiki/Fisher%27s_exact_test#Example
          xsq, p = stats.combine_pvalues([.01, .2, .3], method='fisher')
          assert_approx_equal(p, 0.02156, significant=4)
  
      def test_stouffer(self):
          Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer')
          assert_approx_equal(p, 0.01651, significant=4)
  
      def test_stouffer2(self):
          Z, p = stats.combine_pvalues([.5, .5, .5], method='stouffer')
          assert_approx_equal(p, 0.5, significant=4)
  
      def test_weighted_stouffer(self):
          Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer',
                                       weights=np.ones(3))
          assert_approx_equal(p, 0.01651, significant=4)
  
      def test_weighted_stouffer2(self):
          Z, p = stats.combine_pvalues([.01, .2, .3], method='stouffer',
                                       weights=np.array((1, 4, 9)))
          assert_approx_equal(p, 0.1464, significant=4)
  
      def test_pearson(self):
          Z, p = stats.combine_pvalues([.01, .2, .3], method='pearson')
          assert_approx_equal(p, 0.02213, significant=4)
  
      def test_tippett(self):
          Z, p = stats.combine_pvalues([.01, .2, .3], method='tippett')
          assert_approx_equal(p, 0.0297, significant=4)
  
      def test_mudholkar_george(self):
          Z, p = stats.combine_pvalues([.1, .1, .1], method='mudholkar_george')
          assert_approx_equal(p, 0.019462, significant=4)
  
      def test_mudholkar_george_equal_fisher_pearson_average(self):
          Z, p = stats.combine_pvalues([.01, .2, .3], method='mudholkar_george')
          Z_f, p_f = stats.combine_pvalues([.01, .2, .3], method='fisher')
          Z_p, p_p = stats.combine_pvalues([.01, .2, .3], method='pearson')
          assert_approx_equal(0.5 * (Z_f+Z_p), Z, significant=4)
  
      @pytest.mark.parametrize("variant", ["single", "all", "random"])
      @pytest.mark.parametrize(
          "method",
          ["fisher", "pearson", "tippett", "stouffer", "mudholkar_george"],
      )
      def test_monotonicity(self, variant, method):
          # Test that result increases monotonically with respect to input.
          m, n = 10, 7
          rng = np.random.default_rng(278448169958891062669391462690811630763)
  
          # `pvaluess` is an m × n array of p values. Each row corresponds to
          # a set of p values to be combined with p values increasing
          # monotonically down one column (single), simultaneously down each
          # column (all), or independently down each column (random).
          if variant == "single":
              pvaluess = np.full((m, n), rng.random(n))
              pvaluess[:, 0] = np.linspace(0.1, 0.9, m)
          elif variant == "all":
              pvaluess = np.full((n, m), np.linspace(0.1, 0.9, m)).T
          elif variant == "random":
              pvaluess = np.sort(rng.uniform(0, 1, size=(m, n)), axis=0)
  
          combined_pvalues = [
              stats.combine_pvalues(pvalues, method=method)[1]
              for pvalues in pvaluess
          ]
          assert np.all(np.diff(combined_pvalues) >= 0)
  
  class TestCdfDistanceValidation:
      """
      Test that _cdf_distance() (via wasserstein_distance()) raises ValueErrors
      for bad inputs.
      """
  
      def test_distinct_value_and_weight_lengths(self):
          # When the number of weights does not match the number of values,
          # a ValueError should be raised.
          assert_raises(ValueError, stats.wasserstein_distance,
                        [1], [2], [4], [3, 1])
          assert_raises(ValueError, stats.wasserstein_distance, [1], [2], [1, 0])
  
      def test_zero_weight(self):
          # When a distribution is given zero weight, a ValueError should be
          # raised.
          assert_raises(ValueError, stats.wasserstein_distance,
                        [0, 1], [2], [0, 0])
          assert_raises(ValueError, stats.wasserstein_distance,
                        [0, 1], [2], [3, 1], [0])
  
      def test_negative_weights(self):
          # A ValueError should be raised if there are any negative weights.
          assert_raises(ValueError, stats.wasserstein_distance,
                        [0, 1], [2, 2], [1, 1], [3, -1])
  
      def test_empty_distribution(self):
          # A ValueError should be raised when trying to measure the distance
          # between something and nothing.
          assert_raises(ValueError, stats.wasserstein_distance, [], [2, 2])
          assert_raises(ValueError, stats.wasserstein_distance, [1], [])
  
      def test_inf_weight(self):
          # An inf weight is not valid.
          assert_raises(ValueError, stats.wasserstein_distance,
                        [1, 2, 1], [1, 1], [1, np.inf, 1], [1, 1])
  
  
  class TestWassersteinDistance:
      """ Tests for wasserstein_distance() output values.
      """
  
      def test_simple(self):
          # For basic distributions, the value of the Wasserstein distance is
          # straightforward.
          assert_almost_equal(
              stats.wasserstein_distance([0, 1], [0], [1, 1], [1]),
              .5)
          assert_almost_equal(stats.wasserstein_distance(
              [0, 1], [0], [3, 1], [1]),
              .25)
          assert_almost_equal(stats.wasserstein_distance(
              [0, 2], [0], [1, 1], [1]),
              1)
          assert_almost_equal(stats.wasserstein_distance(
              [0, 1, 2], [1, 2, 3]),
              1)
  
      def test_same_distribution(self):
          # Any distribution moved to itself should have a Wasserstein distance of
          # zero.
          assert_equal(stats.wasserstein_distance([1, 2, 3], [2, 1, 3]), 0)
          assert_equal(
              stats.wasserstein_distance([1, 1, 1, 4], [4, 1],
                                         [1, 1, 1, 1], [1, 3]),
              0)
  
      def test_shift(self):
          # If the whole distribution is shifted by x, then the Wasserstein
          # distance should be x.
          assert_almost_equal(stats.wasserstein_distance([0], [1]), 1)
          assert_almost_equal(stats.wasserstein_distance([-5], [5]), 10)
          assert_almost_equal(
              stats.wasserstein_distance([1, 2, 3, 4, 5], [11, 12, 13, 14, 15]),
              10)
          assert_almost_equal(
              stats.wasserstein_distance([4.5, 6.7, 2.1], [4.6, 7, 9.2],
                                         [3, 1, 1], [1, 3, 1]),
              2.5)
  
      def test_combine_weights(self):
          # Assigning a weight w to a value is equivalent to including that value
          # w times in the value array with weight of 1.
          assert_almost_equal(
              stats.wasserstein_distance(
                  [0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
                  [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
              stats.wasserstein_distance([5, 0, 1], [0, 4, 3],
                                         [1, 2, 4], [1, 2, 4]))
  
      def test_collapse(self):
          # Collapsing a distribution to a point distribution at zero is
          # equivalent to taking the average of the absolute values of the values.
          u = np.arange(-10, 30, 0.3)
          v = np.zeros_like(u)
          assert_almost_equal(
              stats.wasserstein_distance(u, v),
              np.mean(np.abs(u)))
  
          u_weights = np.arange(len(u))
          v_weights = u_weights[::-1]
          assert_almost_equal(
              stats.wasserstein_distance(u, v, u_weights, v_weights),
              np.average(np.abs(u), weights=u_weights))
  
      def test_zero_weight(self):
          # Values with zero weight have no impact on the Wasserstein distance.
          assert_almost_equal(
              stats.wasserstein_distance([1, 2, 100000], [1, 1],
                                         [1, 1, 0], [1, 1]),
              stats.wasserstein_distance([1, 2], [1, 1], [1, 1], [1, 1]))
  
      def test_inf_values(self):
          # Inf values can lead to an inf distance or trigger a RuntimeWarning
          # (and return NaN) if the distance is undefined.
          assert_equal(
              stats.wasserstein_distance([1, 2, np.inf], [1, 1]),
              np.inf)
          assert_equal(
              stats.wasserstein_distance([1, 2, np.inf], [-np.inf, 1]),
              np.inf)
          assert_equal(
              stats.wasserstein_distance([1, -np.inf, np.inf], [1, 1]),
              np.inf)
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "invalid value*")
              assert_equal(
                  stats.wasserstein_distance([1, 2, np.inf], [np.inf, 1]),
                  np.nan)
  
  
  class TestEnergyDistance:
      """ Tests for energy_distance() output values.
      """
  
      def test_simple(self):
          # For basic distributions, the value of the energy distance is
          # straightforward.
          assert_almost_equal(
              stats.energy_distance([0, 1], [0], [1, 1], [1]),
              np.sqrt(2) * .5)
          assert_almost_equal(stats.energy_distance(
              [0, 1], [0], [3, 1], [1]),
              np.sqrt(2) * .25)
          assert_almost_equal(stats.energy_distance(
              [0, 2], [0], [1, 1], [1]),
              2 * .5)
          assert_almost_equal(
              stats.energy_distance([0, 1, 2], [1, 2, 3]),
              np.sqrt(2) * (3*(1./3**2))**.5)
  
      def test_same_distribution(self):
          # Any distribution moved to itself should have a energy distance of
          # zero.
          assert_equal(stats.energy_distance([1, 2, 3], [2, 1, 3]), 0)
          assert_equal(
              stats.energy_distance([1, 1, 1, 4], [4, 1], [1, 1, 1, 1], [1, 3]),
              0)
  
      def test_shift(self):
          # If a single-point distribution is shifted by x, then the energy
          # distance should be sqrt(2) * sqrt(x).
          assert_almost_equal(stats.energy_distance([0], [1]), np.sqrt(2))
          assert_almost_equal(
              stats.energy_distance([-5], [5]),
              np.sqrt(2) * 10**.5)
  
      def test_combine_weights(self):
          # Assigning a weight w to a value is equivalent to including that value
          # w times in the value array with weight of 1.
          assert_almost_equal(
              stats.energy_distance([0, 0, 1, 1, 1, 1, 5], [0, 3, 3, 3, 3, 4, 4],
                                    [1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]),
              stats.energy_distance([5, 0, 1], [0, 4, 3], [1, 2, 4], [1, 2, 4]))
  
      def test_zero_weight(self):
          # Values with zero weight have no impact on the energy distance.
          assert_almost_equal(
              stats.energy_distance([1, 2, 100000], [1, 1], [1, 1, 0], [1, 1]),
              stats.energy_distance([1, 2], [1, 1], [1, 1], [1, 1]))
  
      def test_inf_values(self):
          # Inf values can lead to an inf distance or trigger a RuntimeWarning
          # (and return NaN) if the distance is undefined.
          assert_equal(stats.energy_distance([1, 2, np.inf], [1, 1]), np.inf)
          assert_equal(
              stats.energy_distance([1, 2, np.inf], [-np.inf, 1]),
              np.inf)
          assert_equal(
              stats.energy_distance([1, -np.inf, np.inf], [1, 1]),
              np.inf)
          with suppress_warnings() as sup:
              sup.record(RuntimeWarning, "invalid value*")
              assert_equal(
                  stats.energy_distance([1, 2, np.inf], [np.inf, 1]),
                  np.nan)
  
  
  class TestBrunnerMunzel:
      # Data from (Lumley, 1996)
      X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1]
      Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
      significant = 13
  
      def test_brunnermunzel_one_sided(self):
          # Results are compared with R's lawstat package.
          u1, p1 = stats.brunnermunzel(self.X, self.Y, alternative='less')
          u2, p2 = stats.brunnermunzel(self.Y, self.X, alternative='greater')
          u3, p3 = stats.brunnermunzel(self.X, self.Y, alternative='greater')
          u4, p4 = stats.brunnermunzel(self.Y, self.X, alternative='less')
  
          assert_approx_equal(p1, p2, significant=self.significant)
          assert_approx_equal(p3, p4, significant=self.significant)
          assert_(p1 != p3)
          assert_approx_equal(u1, 3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u2, -3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u3, 3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u4, -3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(p1, 0.0028931043330757342,
                              significant=self.significant)
          assert_approx_equal(p3, 0.99710689566692423,
                              significant=self.significant)
  
      def test_brunnermunzel_two_sided(self):
          # Results are compared with R's lawstat package.
          u1, p1 = stats.brunnermunzel(self.X, self.Y, alternative='two-sided')
          u2, p2 = stats.brunnermunzel(self.Y, self.X, alternative='two-sided')
  
          assert_approx_equal(p1, p2, significant=self.significant)
          assert_approx_equal(u1, 3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u2, -3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(p1, 0.0057862086661515377,
                              significant=self.significant)
  
      def test_brunnermunzel_default(self):
          # The default value for alternative is two-sided
          u1, p1 = stats.brunnermunzel(self.X, self.Y)
          u2, p2 = stats.brunnermunzel(self.Y, self.X)
  
          assert_approx_equal(p1, p2, significant=self.significant)
          assert_approx_equal(u1, 3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u2, -3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(p1, 0.0057862086661515377,
                              significant=self.significant)
  
      def test_brunnermunzel_alternative_error(self):
          alternative = "error"
          distribution = "t"
          nan_policy = "propagate"
          assert_(alternative not in ["two-sided", "greater", "less"])
          assert_raises(ValueError,
                        stats.brunnermunzel,
                        self.X,
                        self.Y,
                        alternative,
                        distribution,
                        nan_policy)
  
      def test_brunnermunzel_distribution_norm(self):
          u1, p1 = stats.brunnermunzel(self.X, self.Y, distribution="normal")
          u2, p2 = stats.brunnermunzel(self.Y, self.X, distribution="normal")
          assert_approx_equal(p1, p2, significant=self.significant)
          assert_approx_equal(u1, 3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u2, -3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(p1, 0.0017041417600383024,
                              significant=self.significant)
  
      def test_brunnermunzel_distribution_error(self):
          alternative = "two-sided"
          distribution = "error"
          nan_policy = "propagate"
          assert_(alternative not in ["t", "normal"])
          assert_raises(ValueError,
                        stats.brunnermunzel,
                        self.X,
                        self.Y,
                        alternative,
                        distribution,
                        nan_policy)
  
      def test_brunnermunzel_empty_imput(self):
          u1, p1 = stats.brunnermunzel(self.X, [])
          u2, p2 = stats.brunnermunzel([], self.Y)
          u3, p3 = stats.brunnermunzel([], [])
  
          assert_equal(u1, np.nan)
          assert_equal(p1, np.nan)
          assert_equal(u2, np.nan)
          assert_equal(p2, np.nan)
          assert_equal(u3, np.nan)
          assert_equal(p3, np.nan)
  
      def test_brunnermunzel_nan_input_propagate(self):
          X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan]
          Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
          u1, p1 = stats.brunnermunzel(X, Y, nan_policy="propagate")
          u2, p2 = stats.brunnermunzel(Y, X, nan_policy="propagate")
  
          assert_equal(u1, np.nan)
          assert_equal(p1, np.nan)
          assert_equal(u2, np.nan)
          assert_equal(p2, np.nan)
  
      def test_brunnermunzel_nan_input_raise(self):
          X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan]
          Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
          alternative = "two-sided"
          distribution = "t"
          nan_policy = "raise"
  
          assert_raises(ValueError,
                        stats.brunnermunzel,
                        X,
                        Y,
                        alternative,
                        distribution,
                        nan_policy)
          assert_raises(ValueError,
                        stats.brunnermunzel,
                        Y,
                        X,
                        alternative,
                        distribution,
                        nan_policy)
  
      def test_brunnermunzel_nan_input_omit(self):
          X = [1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, np.nan]
          Y = [3, 3, 4, 3, 1, 2, 3, 1, 1, 5, 4]
          u1, p1 = stats.brunnermunzel(X, Y, nan_policy="omit")
          u2, p2 = stats.brunnermunzel(Y, X, nan_policy="omit")
  
          assert_approx_equal(p1, p2, significant=self.significant)
          assert_approx_equal(u1, 3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(u2, -3.1374674823029505,
                              significant=self.significant)
          assert_approx_equal(p1, 0.0057862086661515377,
                              significant=self.significant)
  
      def test_brunnermunzel_return_nan(self):
          """ tests that a warning is emitted when p is nan
          p-value with t-distributions can be nan (0/0) (see gh-15843)
          """
          x = [1, 2, 3]
          y = [5, 6, 7, 8, 9]
  
          with pytest.warns(RuntimeWarning, match='p-value cannot be estimated'):
              stats.brunnermunzel(x, y, distribution="t")
  
      def test_brunnermunzel_normal_dist(self):
          """ tests that a p is 0 for datasets that cause p->nan
          when t-distribution is used (see gh-15843)
          """
          x = [1, 2, 3]
          y = [5, 6, 7, 8, 9]
  
          with pytest.warns(RuntimeWarning, match='divide by zero'):
              _, p = stats.brunnermunzel(x, y, distribution="normal")
          assert_equal(p, 0)
  
  
  class TestRatioUniforms:
      """ Tests for rvs_ratio_uniforms.
      """
  
      def test_rv_generation(self):
          # use KS test to check distribution of rvs
          # normal distribution
          f = stats.norm.pdf
          v_bound = np.sqrt(f(np.sqrt(2))) * np.sqrt(2)
          umax, vmin, vmax = np.sqrt(f(0)), -v_bound, v_bound
          rvs = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=2500,
                                         random_state=12345)
          assert_equal(stats.kstest(rvs, 'norm')[1] > 0.25, True)
  
          # exponential distribution
          rvs = stats.rvs_ratio_uniforms(lambda x: np.exp(-x), umax=1,
                                         vmin=0, vmax=2*np.exp(-1),
                                         size=1000, random_state=12345)
          assert_equal(stats.kstest(rvs, 'expon')[1] > 0.25, True)
  
      def test_shape(self):
          # test shape of return value depending on size parameter
          f = stats.norm.pdf
          v_bound = np.sqrt(f(np.sqrt(2))) * np.sqrt(2)
          umax, vmin, vmax = np.sqrt(f(0)), -v_bound, v_bound
  
          r1 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=3,
                                        random_state=1234)
          r2 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=(3,),
                                        random_state=1234)
          r3 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=(3, 1),
                                        random_state=1234)
          assert_equal(r1, r2)
          assert_equal(r2, r3.flatten())
          assert_equal(r1.shape, (3,))
          assert_equal(r3.shape, (3, 1))
  
          r4 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=(3, 3, 3),
                                        random_state=12)
          r5 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=27,
                                        random_state=12)
          assert_equal(r4.flatten(), r5)
          assert_equal(r4.shape, (3, 3, 3))
  
          r6 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, random_state=1234)
          r7 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=1,
                                        random_state=1234)
          r8 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=(1, ),
                                        random_state=1234)
          assert_equal(r6, r7)
          assert_equal(r7, r8)
  
      def test_random_state(self):
          f = stats.norm.pdf
          v_bound = np.sqrt(f(np.sqrt(2))) * np.sqrt(2)
          umax, vmin, vmax = np.sqrt(f(0)), -v_bound, v_bound
          np.random.seed(1234)
          r1 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=(3, 4))
          r2 = stats.rvs_ratio_uniforms(f, umax, vmin, vmax, size=(3, 4),
                                        random_state=1234)
          assert_equal(r1, r2)
  
      def test_exceptions(self):
          f = stats.norm.pdf
          # need vmin < vmax
          assert_raises(ValueError,
                        stats.rvs_ratio_uniforms, pdf=f, umax=1, vmin=3, vmax=1)
          assert_raises(ValueError,
                        stats.rvs_ratio_uniforms, pdf=f, umax=1, vmin=1, vmax=1)
          # need umax > 0
          assert_raises(ValueError,
                        stats.rvs_ratio_uniforms, pdf=f, umax=-1, vmin=1, vmax=1)
          assert_raises(ValueError,
                        stats.rvs_ratio_uniforms, pdf=f, umax=0, vmin=1, vmax=1)
  
  
  class TestMGCErrorWarnings:
      """ Tests errors and warnings derived from MGC.
      """
      def test_error_notndarray(self):
          # raises error if x or y is not a ndarray
          x = np.arange(20)
          y = [5] * 20
          assert_raises(ValueError, stats.multiscale_graphcorr, x, y)
          assert_raises(ValueError, stats.multiscale_graphcorr, y, x)
  
      def test_error_shape(self):
          # raises error if number of samples different (n)
          x = np.arange(100).reshape(25, 4)
          y = x.reshape(10, 10)
          assert_raises(ValueError, stats.multiscale_graphcorr, x, y)
  
      def test_error_lowsamples(self):
          # raises error if samples are low (< 3)
          x = np.arange(3)
          y = np.arange(3)
          assert_raises(ValueError, stats.multiscale_graphcorr, x, y)
  
      def test_error_nans(self):
          # raises error if inputs contain NaNs
          x = np.arange(20, dtype=float)
          x[0] = np.nan
          assert_raises(ValueError, stats.multiscale_graphcorr, x, x)
  
          y = np.arange(20)
          assert_raises(ValueError, stats.multiscale_graphcorr, x, y)
  
      def test_error_wrongdisttype(self):
          # raises error if metric is not a function
          x = np.arange(20)
          compute_distance = 0
          assert_raises(ValueError, stats.multiscale_graphcorr, x, x,
                        compute_distance=compute_distance)
  
      @pytest.mark.parametrize("reps", [
          -1,    # reps is negative
          '1',   # reps is not integer
      ])
      def test_error_reps(self, reps):
          # raises error if reps is negative
          x = np.arange(20)
          assert_raises(ValueError, stats.multiscale_graphcorr, x, x, reps=reps)
  
      def test_warns_reps(self):
          # raises warning when reps is less than 1000
          x = np.arange(20)
          reps = 100
          assert_warns(RuntimeWarning, stats.multiscale_graphcorr, x, x, reps=reps)
  
      def test_error_infty(self):
          # raises error if input contains infinities
          x = np.arange(20)
          y = np.ones(20) * np.inf
          assert_raises(ValueError, stats.multiscale_graphcorr, x, y)
  
  
  class TestMGCStat:
      """ Test validity of MGC test statistic
      """
      def _simulations(self, samps=100, dims=1, sim_type=""):
          # linear simulation
          if sim_type == "linear":
              x = np.random.uniform(-1, 1, size=(samps, 1))
              y = x + 0.3 * np.random.random_sample(size=(x.size, 1))
  
          # spiral simulation
          elif sim_type == "nonlinear":
              unif = np.array(np.random.uniform(0, 5, size=(samps, 1)))
              x = unif * np.cos(np.pi * unif)
              y = unif * np.sin(np.pi * unif) + (0.4
                  * np.random.random_sample(size=(x.size, 1)))
  
          # independence (tests type I simulation)
          elif sim_type == "independence":
              u = np.random.normal(0, 1, size=(samps, 1))
              v = np.random.normal(0, 1, size=(samps, 1))
              u_2 = np.random.binomial(1, p=0.5, size=(samps, 1))
              v_2 = np.random.binomial(1, p=0.5, size=(samps, 1))
              x = u/3 + 2*u_2 - 1
              y = v/3 + 2*v_2 - 1
  
          # raises error if not approved sim_type
          else:
              raise ValueError("sim_type must be linear, nonlinear, or "
                               "independence")
  
          # add dimensions of noise for higher dimensions
          if dims > 1:
              dims_noise = np.random.normal(0, 1, size=(samps, dims-1))
              x = np.concatenate((x, dims_noise), axis=1)
  
          return x, y
  
      @pytest.mark.slow
      @pytest.mark.parametrize("sim_type, obs_stat, obs_pvalue", [
          ("linear", 0.97, 1/1000),           # test linear simulation
          ("nonlinear", 0.163, 1/1000),       # test spiral simulation
          ("independence", -0.0094, 0.78)     # test independence simulation
      ])
      def test_oned(self, sim_type, obs_stat, obs_pvalue):
          np.random.seed(12345678)
  
          # generate x and y
          x, y = self._simulations(samps=100, dims=1, sim_type=sim_type)
  
          # test stat and pvalue
          stat, pvalue, _ = stats.multiscale_graphcorr(x, y)
          assert_approx_equal(stat, obs_stat, significant=1)
          assert_approx_equal(pvalue, obs_pvalue, significant=1)
  
      @pytest.mark.slow
      @pytest.mark.parametrize("sim_type, obs_stat, obs_pvalue", [
          ("linear", 0.184, 1/1000),           # test linear simulation
          ("nonlinear", 0.0190, 0.117),        # test spiral simulation
      ])
      def test_fived(self, sim_type, obs_stat, obs_pvalue):
          np.random.seed(12345678)
  
          # generate x and y
          x, y = self._simulations(samps=100, dims=5, sim_type=sim_type)
  
          # test stat and pvalue
          stat, pvalue, _ = stats.multiscale_graphcorr(x, y)
          assert_approx_equal(stat, obs_stat, significant=1)
          assert_approx_equal(pvalue, obs_pvalue, significant=1)
  
      @pytest.mark.xslow
      def test_twosamp(self):
          np.random.seed(12345678)
  
          # generate x and y
          x = np.random.binomial(100, 0.5, size=(100, 5))
          y = np.random.normal(0, 1, size=(80, 5))
  
          # test stat and pvalue
          stat, pvalue, _ = stats.multiscale_graphcorr(x, y)
          assert_approx_equal(stat, 1.0, significant=1)
          assert_approx_equal(pvalue, 0.001, significant=1)
  
          # generate x and y
          y = np.random.normal(0, 1, size=(100, 5))
  
          # test stat and pvalue
          stat, pvalue, _ = stats.multiscale_graphcorr(x, y, is_twosamp=True)
          assert_approx_equal(stat, 1.0, significant=1)
          assert_approx_equal(pvalue, 0.001, significant=1)
  
      @pytest.mark.slow
      def test_workers(self):
          np.random.seed(12345678)
  
          # generate x and y
          x, y = self._simulations(samps=100, dims=1, sim_type="linear")
  
          # test stat and pvalue
          stat, pvalue, _ = stats.multiscale_graphcorr(x, y, workers=2)
          assert_approx_equal(stat, 0.97, significant=1)
          assert_approx_equal(pvalue, 0.001, significant=1)
  
      @pytest.mark.slow
      def test_random_state(self):
          # generate x and y
          x, y = self._simulations(samps=100, dims=1, sim_type="linear")
  
          # test stat and pvalue
          stat, pvalue, _ = stats.multiscale_graphcorr(x, y, random_state=1)
          assert_approx_equal(stat, 0.97, significant=1)
          assert_approx_equal(pvalue, 0.001, significant=1)
  
      @pytest.mark.slow
      def test_dist_perm(self):
          np.random.seed(12345678)
          # generate x and y
          x, y = self._simulations(samps=100, dims=1, sim_type="nonlinear")
          distx = cdist(x, x, metric="euclidean")
          disty = cdist(y, y, metric="euclidean")
  
          stat_dist, pvalue_dist, _ = stats.multiscale_graphcorr(distx, disty,
                                                                 compute_distance=None,
                                                                 random_state=1)
          assert_approx_equal(stat_dist, 0.163, significant=1)
          assert_approx_equal(pvalue_dist, 0.001, significant=1)
  
      @pytest.mark.slow
      def test_pvalue_literature(self):
          np.random.seed(12345678)
  
          # generate x and y
          x, y = self._simulations(samps=100, dims=1, sim_type="linear")
  
          # test stat and pvalue
          _, pvalue, _ = stats.multiscale_graphcorr(x, y, random_state=1)
          assert_allclose(pvalue, 1/1001)
  
  
  class TestPageTrendTest:
      # expected statistic and p-values generated using R at
      # https://rdrr.io/cran/cultevo/, e.g.
      # library(cultevo)
      # data = rbind(c(72, 47, 73, 35, 47, 96, 30, 59, 41, 36, 56, 49, 81, 43,
      #                   70, 47, 28, 28, 62, 20, 61, 20, 80, 24, 50),
      #              c(68, 52, 60, 34, 44, 20, 65, 88, 21, 81, 48, 31, 31, 67,
      #                69, 94, 30, 24, 40, 87, 70, 43, 50, 96, 43),
      #              c(81, 13, 85, 35, 79, 12, 92, 86, 21, 64, 16, 64, 68, 17,
      #                16, 89, 71, 43, 43, 36, 54, 13, 66, 51, 55))
      # result = page.test(data, verbose=FALSE)
      # Most test cases generated to achieve common critical p-values so that
      # results could be checked (to limited precision) against tables in
      # scipy.stats.page_trend_test reference [1]
  
      np.random.seed(0)
      data_3_25 = np.random.rand(3, 25)
      data_10_26 = np.random.rand(10, 26)
  
      ts = [
            (12805, 0.3886487053947608, False, 'asymptotic', data_3_25),
            (49140, 0.02888978556179862, False, 'asymptotic', data_10_26),
            (12332, 0.7722477197436702, False, 'asymptotic',
             [[72, 47, 73, 35, 47, 96, 30, 59, 41, 36, 56, 49, 81,
               43, 70, 47, 28, 28, 62, 20, 61, 20, 80, 24, 50],
              [68, 52, 60, 34, 44, 20, 65, 88, 21, 81, 48, 31, 31,
               67, 69, 94, 30, 24, 40, 87, 70, 43, 50, 96, 43],
              [81, 13, 85, 35, 79, 12, 92, 86, 21, 64, 16, 64, 68,
               17, 16, 89, 71, 43, 43, 36, 54, 13, 66, 51, 55]]),
            (266, 4.121656378600823e-05, False, 'exact',
             [[1.5, 4., 8.3, 5, 19, 11],
              [5, 4, 3.5, 10, 20, 21],
              [8.4, 3.2, 10, 12, 14, 15]]),
            (332, 0.9566400920502488, True, 'exact',
             [[4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1],
              [4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1], [4, 3, 2, 1],
              [3, 4, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4],
              [1, 2, 3, 4], [1, 2, 3, 4]]),
            (241, 0.9622210164861476, True, 'exact',
             [[3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
              [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
              [3, 2, 1], [2, 1, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3],
              [1, 2, 3], [1, 2, 3], [1, 2, 3]]),
            (197, 0.9619432897162209, True, 'exact',
             [[6, 5, 4, 3, 2, 1], [6, 5, 4, 3, 2, 1], [1, 3, 4, 5, 2, 6]]),
            (423, 0.9590458306880073, True, 'exact',
             [[5, 4, 3, 2, 1], [5, 4, 3, 2, 1], [5, 4, 3, 2, 1],
              [5, 4, 3, 2, 1], [5, 4, 3, 2, 1], [5, 4, 3, 2, 1],
              [4, 1, 3, 2, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5],
              [1, 2, 3, 4, 5]]),
            (217, 0.9693058575034678, True, 'exact',
             [[3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
              [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
              [2, 1, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3], [1, 2, 3],
              [1, 2, 3]]),
            (395, 0.991530289351305, True, 'exact',
             [[7, 6, 5, 4, 3, 2, 1], [7, 6, 5, 4, 3, 2, 1],
              [6, 5, 7, 4, 3, 2, 1], [1, 2, 3, 4, 5, 6, 7]]),
            (117, 0.9997817843373017, True, 'exact',
             [[3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1], [3, 2, 1],
              [3, 2, 1], [3, 2, 1], [3, 2, 1], [2, 1, 3], [1, 2, 3]]),
           ]
  
      @pytest.mark.parametrize("L, p, ranked, method, data", ts)
      def test_accuracy(self, L, p, ranked, method, data):
          np.random.seed(42)
          res = stats.page_trend_test(data, ranked=ranked, method=method)
          assert_equal(L, res.statistic)
          assert_allclose(p, res.pvalue)
          assert_equal(method, res.method)
  
      ts2 = [
             (542, 0.9481266260876332, True, 'exact',
              [[10, 9, 8, 7, 6, 5, 4, 3, 2, 1],
               [1, 8, 4, 7, 6, 5, 9, 3, 2, 10]]),
             (1322, 0.9993113928199309, True, 'exact',
              [[10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [10, 9, 8, 7, 6, 5, 4, 3, 2, 1],
               [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [9, 2, 8, 7, 6, 5, 4, 3, 10, 1],
               [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]),
             (2286, 0.9908688345484833, True, 'exact',
              [[8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
               [8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
               [8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
               [8, 7, 6, 5, 4, 3, 2, 1], [8, 7, 6, 5, 4, 3, 2, 1],
               [8, 7, 6, 5, 4, 3, 2, 1], [1, 3, 5, 6, 4, 7, 2, 8],
               [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8],
               [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8],
               [1, 2, 3, 4, 5, 6, 7, 8]]),
            ]
  
      # only the first of these appears slow because intermediate data are
      # cached and used on the rest
      @pytest.mark.parametrize("L, p, ranked, method, data", ts)
      @pytest.mark.slow()
      def test_accuracy2(self, L, p, ranked, method, data):
          np.random.seed(42)
          res = stats.page_trend_test(data, ranked=ranked, method=method)
          assert_equal(L, res.statistic)
          assert_allclose(p, res.pvalue)
          assert_equal(method, res.method)
  
      def test_options(self):
          np.random.seed(42)
          m, n = 10, 20
          predicted_ranks = np.arange(1, n+1)
          perm = np.random.permutation(np.arange(n))
          data = np.random.rand(m, n)
          ranks = stats.rankdata(data, axis=1)
          res1 = stats.page_trend_test(ranks)
          res2 = stats.page_trend_test(ranks, ranked=True)
          res3 = stats.page_trend_test(data, ranked=False)
          res4 = stats.page_trend_test(ranks, predicted_ranks=predicted_ranks)
          res5 = stats.page_trend_test(ranks[:, perm],
                                       predicted_ranks=predicted_ranks[perm])
          assert_equal(res1.statistic, res2.statistic)
          assert_equal(res1.statistic, res3.statistic)
          assert_equal(res1.statistic, res4.statistic)
          assert_equal(res1.statistic, res5.statistic)
  
      def test_Ames_assay(self):
          # test from _page_trend_test.py [2] page 151; data on page 144
          np.random.seed(42)
  
          data = [[101, 117, 111], [91, 90, 107], [103, 133, 121],
                  [136, 140, 144], [190, 161, 201], [146, 120, 116]]
          data = np.array(data).T
          predicted_ranks = np.arange(1, 7)
  
          res = stats.page_trend_test(data, ranked=False,
                                      predicted_ranks=predicted_ranks,
                                      method="asymptotic")
          assert_equal(res.statistic, 257)
          assert_almost_equal(res.pvalue, 0.0035, decimal=4)
  
          res = stats.page_trend_test(data, ranked=False,
                                      predicted_ranks=predicted_ranks,
                                      method="exact")
          assert_equal(res.statistic, 257)
          assert_almost_equal(res.pvalue, 0.0023, decimal=4)
  
      def test_input_validation(self):
          # test data not a 2d array
          with assert_raises(ValueError, match="`data` must be a 2d array."):
              stats.page_trend_test(None)
          with assert_raises(ValueError, match="`data` must be a 2d array."):
              stats.page_trend_test([])
          with assert_raises(ValueError, match="`data` must be a 2d array."):
              stats.page_trend_test([1, 2])
          with assert_raises(ValueError, match="`data` must be a 2d array."):
              stats.page_trend_test([[[1]]])
  
          # test invalid dimensions
          with assert_raises(ValueError, match="Page's L is only appropriate"):
              stats.page_trend_test(np.random.rand(1, 3))
          with assert_raises(ValueError, match="Page's L is only appropriate"):
              stats.page_trend_test(np.random.rand(2, 2))
  
          # predicted ranks must include each integer [1, 2, 3] exactly once
          message = "`predicted_ranks` must include each integer"
          with assert_raises(ValueError, match=message):
              stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                    predicted_ranks=[0, 1, 2])
          with assert_raises(ValueError, match=message):
              stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                    predicted_ranks=[1.1, 2, 3])
          with assert_raises(ValueError, match=message):
              stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                    predicted_ranks=[1, 2, 3, 3])
          with assert_raises(ValueError, match=message):
              stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                    predicted_ranks="invalid")
  
          # test improperly ranked data
          with assert_raises(ValueError, match="`data` is not properly ranked"):
              stats.page_trend_test([[0, 2, 3], [1, 2, 3]], True)
          with assert_raises(ValueError, match="`data` is not properly ranked"):
              stats.page_trend_test([[1, 2, 3], [1, 2, 4]], True)
  
          # various
          with assert_raises(ValueError, match="`data` contains NaNs"):
              stats.page_trend_test([[1, 2, 3], [1, 2, np.nan]],
                                    ranked=False)
          with assert_raises(ValueError, match="`method` must be in"):
              stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                    method="ekki")
          with assert_raises(TypeError, match="`ranked` must be boolean."):
              stats.page_trend_test(data=[[1, 2, 3], [1, 2, 3]],
                                    ranked="ekki")
  
  
  rng = np.random.default_rng(902340982)
  x = rng.random(10)
  y = rng.random(10)
  
  
  @pytest.mark.parametrize("fun, args",
                           [(stats.wilcoxon, (x,)),
                            (stats.ks_1samp, (x, stats.norm.cdf)),  # type: ignore[attr-defined] # noqa
                            (stats.ks_2samp, (x, y)),
                            (stats.kstest, (x, y)),
                            ])
  def test_rename_mode_method(fun, args):
  
      res = fun(*args, method='exact')
      res2 = fun(*args, mode='exact')
      assert_equal(res, res2)
  
      err = rf"{fun.__name__}() got multiple values for argument"
      with pytest.raises(TypeError, match=re.escape(err)):
          fun(*args, method='exact', mode='exact')
  