content: # Author:  Travis Oliphant, 2002
  #
  # Further enhancements and tests added by numerous SciPy developers.
  #
  from __future__ import division, print_function, absolute_import
  
  import warnings
  
  import numpy as np
  from numpy.random import RandomState
  from numpy.testing import (TestCase, run_module_suite, assert_array_equal,
      assert_almost_equal, assert_array_less, assert_array_almost_equal,
      assert_raises, assert_, assert_allclose, assert_equal, dec, assert_warns)
  
  from scipy import stats
  
  # Matplotlib is not a scipy dependency but is optionally used in probplot, so
  # check if it's available
  try:
      import matplotlib.pyplot as plt
      have_matplotlib = True
  except:
      have_matplotlib = False
  
  
  g1 = [1.006, 0.996, 0.998, 1.000, 0.992, 0.993, 1.002, 0.999, 0.994, 1.000]
  g2 = [0.998, 1.006, 1.000, 1.002, 0.997, 0.998, 0.996, 1.000, 1.006, 0.988]
  g3 = [0.991, 0.987, 0.997, 0.999, 0.995, 0.994, 1.000, 0.999, 0.996, 0.996]
  g4 = [1.005, 1.002, 0.994, 1.000, 0.995, 0.994, 0.998, 0.996, 1.002, 0.996]
  g5 = [0.998, 0.998, 0.982, 0.990, 1.002, 0.984, 0.996, 0.993, 0.980, 0.996]
  g6 = [1.009, 1.013, 1.009, 0.997, 0.988, 1.002, 0.995, 0.998, 0.981, 0.996]
  g7 = [0.990, 1.004, 0.996, 1.001, 0.998, 1.000, 1.018, 1.010, 0.996, 1.002]
  g8 = [0.998, 1.000, 1.006, 1.000, 1.002, 0.996, 0.998, 0.996, 1.002, 1.006]
  g9 = [1.002, 0.998, 0.996, 0.995, 0.996, 1.004, 1.004, 0.998, 0.999, 0.991]
  g10 = [0.991, 0.995, 0.984, 0.994, 0.997, 0.997, 0.991, 0.998, 1.004, 0.997]
  
  
  class TestShapiro(TestCase):
      def test_basic(self):
          x1 = [0.11,7.87,4.61,10.14,7.95,3.14,0.46,
                4.43,0.21,4.75,0.71,1.52,3.24,
                0.93,0.42,4.97,9.53,4.55,0.47,6.66]
          w,pw = stats.shapiro(x1)
          assert_almost_equal(w,0.90047299861907959,6)
          assert_almost_equal(pw,0.042089745402336121,6)
          x2 = [1.36,1.14,2.92,2.55,1.46,1.06,5.27,-1.11,
                3.48,1.10,0.88,-0.51,1.46,0.52,6.20,1.69,
                0.08,3.67,2.81,3.49]
          w,pw = stats.shapiro(x2)
          assert_almost_equal(w,0.9590270,6)
          assert_almost_equal(pw,0.52460,3)
  
      def test_bad_arg(self):
          # Length of x is less than 3.
          x = [1]
          assert_raises(ValueError, stats.shapiro, x)
  
  
  class TestAnderson(TestCase):
      def test_normal(self):
          rs = RandomState(1234567890)
          x1 = rs.standard_exponential(size=50)
          x2 = rs.standard_normal(size=50)
          A,crit,sig = stats.anderson(x1)
          assert_array_less(crit[:-1], A)
          A,crit,sig = stats.anderson(x2)
          assert_array_less(A, crit[-2:])
  
      def test_expon(self):
          rs = RandomState(1234567890)
          x1 = rs.standard_exponential(size=50)
          x2 = rs.standard_normal(size=50)
          A,crit,sig = stats.anderson(x1,'expon')
          assert_array_less(A, crit[-2:])
          olderr = np.seterr(all='ignore')
          try:
              A,crit,sig = stats.anderson(x2,'expon')
          finally:
              np.seterr(**olderr)
          assert_(A > crit[-1])
  
      def test_bad_arg(self):
          assert_raises(ValueError, stats.anderson, [1], dist='plate_of_shrimp')
  
  
  class TestAndersonKSamp(TestCase):
      def test_example1a(self):
          # Example data from Scholz & Stephens (1987), originally
          # published in Lehmann (1995, Nonparametrics, Statistical
          # Methods Based on Ranks, p. 309)
          # Pass a mixture of lists and arrays
          t1 = [38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0]
          t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
          t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
          t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])
          assert_warns(UserWarning, stats.anderson_ksamp, (t1, t2, t3, t4),
                       midrank=False)
          with warnings.catch_warnings():
              warnings.filterwarnings('ignore', message='approximate p-value')
              Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4), midrank=False)
  
          assert_almost_equal(Tk, 4.449, 3)
          assert_array_almost_equal([0.4985, 1.3237, 1.9158, 2.4930, 3.2459],
                                    tm, 4)
          assert_almost_equal(p, 0.0021, 4)
  
      def test_example1b(self):
          # Example data from Scholz & Stephens (1987), originally
          # published in Lehmann (1995, Nonparametrics, Statistical
          # Methods Based on Ranks, p. 309)
          # Pass arrays
          t1 = np.array([38.7, 41.5, 43.8, 44.5, 45.5, 46.0, 47.7, 58.0])
          t2 = np.array([39.2, 39.3, 39.7, 41.4, 41.8, 42.9, 43.3, 45.8])
          t3 = np.array([34.0, 35.0, 39.0, 40.0, 43.0, 43.0, 44.0, 45.0])
          t4 = np.array([34.0, 34.8, 34.8, 35.4, 37.2, 37.8, 41.2, 42.8])
          with warnings.catch_warnings():
              warnings.filterwarnings('ignore', message='approximate p-value')
              Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4), midrank=True)
  
          assert_almost_equal(Tk, 4.480, 3)
          assert_array_almost_equal([0.4985, 1.3237, 1.9158, 2.4930, 3.2459],
                                    tm, 4)
          assert_almost_equal(p, 0.0020, 4)
  
      def test_example2a(self):
          # Example data taken from an earlier technical report of
          # Scholz and Stephens
          # Pass lists instead of arrays
          t1 = [194, 15, 41, 29, 33, 181]
          t2 = [413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118]
          t3 = [34, 31, 18, 18, 67, 57, 62, 7, 22, 34]
          t4 = [90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29,
                118, 25, 156, 310, 76, 26, 44, 23, 62]
          t5 = [130, 208, 70, 101, 208]
          t6 = [74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27]
          t7 = [55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33]
          t8 = [23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5,
                12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95]
          t9 = [97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82,
                54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24]
          t10 = [50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36,
                 22, 139, 210, 97, 30, 23, 13, 14]
          t11 = [359, 9, 12, 270, 603, 3, 104, 2, 438]
          t12 = [50, 254, 5, 283, 35, 12]
          t13 = [487, 18, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130]
          t14 = [102, 209, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 230, 66,
                 61, 34]
          with warnings.catch_warnings():
              warnings.filterwarnings('ignore', message='approximate p-value')
              Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4, t5, t6, t7, t8,
                                                t9, t10, t11, t12, t13, t14),
                                               midrank=False)
  
          assert_almost_equal(Tk, 3.288, 3)
          assert_array_almost_equal([0.5990, 1.3269, 1.8052, 2.2486, 2.8009],
                                    tm, 4)
          assert_almost_equal(p, 0.0041, 4)
  
      def test_example2b(self):
          # Example data taken from an earlier technical report of
          # Scholz and Stephens
          t1 = [194, 15, 41, 29, 33, 181]
          t2 = [413, 14, 58, 37, 100, 65, 9, 169, 447, 184, 36, 201, 118]
          t3 = [34, 31, 18, 18, 67, 57, 62, 7, 22, 34]
          t4 = [90, 10, 60, 186, 61, 49, 14, 24, 56, 20, 79, 84, 44, 59, 29,
                118, 25, 156, 310, 76, 26, 44, 23, 62]
          t5 = [130, 208, 70, 101, 208]
          t6 = [74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27]
          t7 = [55, 320, 56, 104, 220, 239, 47, 246, 176, 182, 33]
          t8 = [23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5,
                12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95]
          t9 = [97, 51, 11, 4, 141, 18, 142, 68, 77, 80, 1, 16, 106, 206, 82,
                54, 31, 216, 46, 111, 39, 63, 18, 191, 18, 163, 24]
          t10 = [50, 44, 102, 72, 22, 39, 3, 15, 197, 188, 79, 88, 46, 5, 5, 36,
                 22, 139, 210, 97, 30, 23, 13, 14]
          t11 = [359, 9, 12, 270, 603, 3, 104, 2, 438]
          t12 = [50, 254, 5, 283, 35, 12]
          t13 = [487, 18, 100, 7, 98, 5, 85, 91, 43, 230, 3, 130]
          t14 = [102, 209, 14, 57, 54, 32, 67, 59, 134, 152, 27, 14, 230, 66,
                 61, 34]
          with warnings.catch_warnings():
              warnings.filterwarnings('ignore', message='approximate p-value')
              Tk, tm, p = stats.anderson_ksamp((t1, t2, t3, t4, t5, t6, t7, t8,
                                                t9, t10, t11, t12, t13, t14),
                                               midrank=True)
  
          assert_almost_equal(Tk, 3.294, 3)
          assert_array_almost_equal([0.5990, 1.3269, 1.8052, 2.2486, 2.8009],
                                    tm, 4)
          assert_almost_equal(p, 0.0041, 4)
  
      def test_not_enough_samples(self):
          assert_raises(ValueError, stats.anderson_ksamp, np.ones(5))
  
      def test_no_distinct_observations(self):
          assert_raises(ValueError, stats.anderson_ksamp,
                        (np.ones(5), np.ones(5)))
  
      def test_empty_sample(self):
          assert_raises(ValueError, stats.anderson_ksamp, (np.ones(5), []))
  
  
  class TestAnsari(TestCase):
  
      def test_small(self):
          x = [1,2,3,3,4]
          y = [3,2,6,1,6,1,4,1]
          W, pval = stats.ansari(x,y)
          assert_almost_equal(W,23.5,11)
          assert_almost_equal(pval,0.13499256881897437,11)
  
      def test_approx(self):
          ramsay = np.array((111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
                             101, 96, 97, 102, 107, 113, 116, 113, 110, 98))
          parekh = np.array((107, 108, 106, 98, 105, 103, 110, 105, 104,
                             100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99))
  
          with warnings.catch_warnings():
              warnings.filterwarnings('ignore',
                          message="Ties preclude use of exact statistic.")
              W, pval = stats.ansari(ramsay, parekh)
  
          assert_almost_equal(W,185.5,11)
          assert_almost_equal(pval,0.18145819972867083,11)
  
      def test_exact(self):
          W,pval = stats.ansari([1,2,3,4],[15,5,20,8,10,12])
          assert_almost_equal(W,10.0,11)
          assert_almost_equal(pval,0.533333333333333333,7)
  
      def test_bad_arg(self):
          assert_raises(ValueError, stats.ansari, [], [1])
          assert_raises(ValueError, stats.ansari, [1], [])
  
  
  class TestBartlett(TestCase):
  
      def test_data(self):
          args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
          T, pval = stats.bartlett(*args)
          assert_almost_equal(T,20.78587342806484,7)
          assert_almost_equal(pval,0.0136358632781,7)
  
      def test_bad_arg(self):
          # Too few args raises ValueError.
          assert_raises(ValueError, stats.bartlett, [1])
  
  
  class TestLevene(TestCase):
  
      def test_data(self):
          args = [g1, g2, g3, g4, g5, g6, g7, g8, g9, g10]
          W, pval = stats.levene(*args)
          assert_almost_equal(W,1.7059176930008939,7)
          assert_almost_equal(pval,0.0990829755522,7)
  
      def test_trimmed1(self):
          # Test that center='trimmed' gives the same result as center='mean'
          # when proportiontocut=0.
          W1, pval1 = stats.levene(g1, g2, g3, center='mean')
          W2, pval2 = stats.levene(g1, g2, g3, center='trimmed', proportiontocut=0.0)
          assert_almost_equal(W1, W2)
          assert_almost_equal(pval1, pval2)
  
      def test_trimmed2(self):
          x = [1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 100.0]
          y = [0.0, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 200.0]
          np.random.seed(1234)
          x2 = np.random.permutation(x)
  
          # Use center='trimmed'
          W0, pval0 = stats.levene(x, y, center='trimmed', proportiontocut=0.125)
          W1, pval1 = stats.levene(x2, y, center='trimmed', proportiontocut=0.125)
          # Trim the data here, and use center='mean'
          W2, pval2 = stats.levene(x[1:-1], y[1:-1], center='mean')
          # Result should be the same.
          assert_almost_equal(W0, W2)
          assert_almost_equal(W1, W2)
          assert_almost_equal(pval1, pval2)
  
      def test_equal_mean_median(self):
          x = np.linspace(-1,1,21)
          np.random.seed(1234)
          x2 = np.random.permutation(x)
          y = x**3
          W1, pval1 = stats.levene(x, y, center='mean')
          W2, pval2 = stats.levene(x2, y, center='median')
          assert_almost_equal(W1, W2)
          assert_almost_equal(pval1, pval2)
  
      def test_bad_keyword(self):
          x = np.linspace(-1,1,21)
          assert_raises(TypeError, stats.levene, x, x, portiontocut=0.1)
  
      def test_bad_center_value(self):
          x = np.linspace(-1,1,21)
          assert_raises(ValueError, stats.levene, x, x, center='trim')
  
      def test_too_few_args(self):
          assert_raises(ValueError, stats.levene, [1])
  
  
  class TestBinomP(TestCase):
  
      def test_data(self):
          pval = stats.binom_test(100,250)
          assert_almost_equal(pval,0.0018833009350757682,11)
          pval = stats.binom_test(201,405)
          assert_almost_equal(pval,0.92085205962670713,11)
          pval = stats.binom_test([682,243],p=3.0/4)
          assert_almost_equal(pval,0.38249155957481695,11)
  
      def test_bad_len_x(self):
          # Length of x must be 1 or 2.
          assert_raises(ValueError, stats.binom_test, [1,2,3])
  
      def test_bad_n(self):
          # len(x) is 1, but n is invalid.
          # Missing n
          assert_raises(ValueError, stats.binom_test, [100])
          # n less than x[0]
          assert_raises(ValueError, stats.binom_test, [100], n=50)
  
      def test_bad_p(self):
          assert_raises(ValueError, stats.binom_test, [50, 50], p=2.0)
  
  
  class TestFindRepeats(TestCase):
  
      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.
          a = [10, 20, 50, 30, 40]
          repeated, counts = stats.find_repeats(a)
          assert_array_equal(repeated, [])
          assert_array_equal(counts, [])
  
  
  class TestFligner(TestCase):
  
      def test_data(self):
          # numbers from R: fligner.test in package stats
          x1 = np.arange(5)
          assert_array_almost_equal(stats.fligner(x1,x1**2),
                             (3.2282229927203536, 0.072379187848207877), 11)
  
      def test_trimmed1(self):
          # Test that center='trimmed' gives the same result as center='mean'
          # when proportiontocut=0.
          Xsq1, pval1 = stats.fligner(g1, g2, g3, center='mean')
          Xsq2, pval2 = stats.fligner(g1, g2, g3, center='trimmed', proportiontocut=0.0)
          assert_almost_equal(Xsq1, Xsq2)
          assert_almost_equal(pval1, pval2)
  
      def test_trimmed2(self):
          x = [1.2, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 100.0]
          y = [0.0, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 200.0]
          # Use center='trimmed'
          Xsq1, pval1 = stats.fligner(x, y, center='trimmed', proportiontocut=0.125)
          # Trim the data here, and use center='mean'
          Xsq2, pval2 = stats.fligner(x[1:-1], y[1:-1], center='mean')
          # Result should be the same.
          assert_almost_equal(Xsq1, Xsq2)
          assert_almost_equal(pval1, pval2)
  
      # The following test looks reasonable at first, but fligner() uses the
      # function stats.rankdata(), and in one of the cases in this test,
      # there are ties, while in the other (because of normal rounding
      # errors) there are not.  This difference leads to differences in the
      # third significant digit of W.
      #
      #def test_equal_mean_median(self):
      #    x = np.linspace(-1,1,21)
      #    y = x**3
      #    W1, pval1 = stats.fligner(x, y, center='mean')
      #    W2, pval2 = stats.fligner(x, y, center='median')
      #    assert_almost_equal(W1, W2)
      #    assert_almost_equal(pval1, pval2)
  
      def test_bad_keyword(self):
          x = np.linspace(-1,1,21)
          assert_raises(TypeError, stats.fligner, x, x, portiontocut=0.1)
  
      def test_bad_center_value(self):
          x = np.linspace(-1,1,21)
          assert_raises(ValueError, stats.fligner, x, x, center='trim')
  
      def test_bad_num_args(self):
          # Too few args raises ValueError.
          assert_raises(ValueError, stats.fligner, [1])
  
  
  class TestMood(TestCase):
      def test_mood(self):
          # numbers from R: mood.test in package stats
          x1 = np.arange(5)
          assert_array_almost_equal(stats.mood(x1, x1**2),
                                    (-1.3830857299399906, 0.16663858066771478), 11)
  
      def test_mood_order_of_args(self):
          # z should change sign when the order of arguments changes, pvalue
          # should not change
          np.random.seed(1234)
          x1 = np.random.randn(10, 1)
          x2 = np.random.randn(15, 1)
          z1, p1 = stats.mood(x1, x2)
          z2, p2 = stats.mood(x2, x1)
          assert_array_almost_equal([z1, p1], [-z2, p2])
  
      def test_mood_with_axis_none(self):
          #Test with axis = None, compare with results from R
          x1 = [-0.626453810742332, 0.183643324222082, -0.835628612410047,
                 1.59528080213779, 0.329507771815361, -0.820468384118015,
                 0.487429052428485, 0.738324705129217, 0.575781351653492,
                -0.305388387156356, 1.51178116845085, 0.389843236411431,
                -0.621240580541804, -2.2146998871775, 1.12493091814311,
                -0.0449336090152309, -0.0161902630989461, 0.943836210685299,
                 0.821221195098089, 0.593901321217509]
  
          x2 = [-0.896914546624981, 0.184849184646742, 1.58784533120882,
                -1.13037567424629, -0.0802517565509893, 0.132420284381094,
                 0.707954729271733, -0.23969802417184, 1.98447393665293,
                -0.138787012119665, 0.417650750792556, 0.981752777463662,
                -0.392695355503813, -1.03966897694891, 1.78222896030858,
                -2.31106908460517, 0.878604580921265, 0.035806718015226,
                 1.01282869212708, 0.432265154539617, 2.09081920524915,
                -1.19992581964387, 1.58963820029007, 1.95465164222325,
                 0.00493777682814261, -2.45170638784613, 0.477237302613617,
                -0.596558168631403, 0.792203270299649, 0.289636710177348]
  
          x1 = np.array(x1)
          x2 = np.array(x2)
          x1.shape = (10, 2)
          x2.shape = (15, 2)
          assert_array_almost_equal(stats.mood(x1, x2, axis=None),
                                    [-1.31716607555, 0.18778296257])
  
      def test_mood_2d(self):
          # Test if the results of mood test in 2-D case are consistent with the
          # R result for the same inputs.  Numbers from R mood.test().
          ny = 5
          np.random.seed(1234)
          x1 = np.random.randn(10, ny)
          x2 = np.random.randn(15, ny)
          z_vectest, pval_vectest = stats.mood(x1, x2)
  
          for j in range(ny):
              assert_array_almost_equal([z_vectest[j], pval_vectest[j]],
                                        stats.mood(x1[:, j], x2[:, j]))
  
          # inverse order of dimensions
          x1 = x1.transpose()
          x2 = x2.transpose()
          z_vectest, pval_vectest = stats.mood(x1, x2, axis=1)
  
          for i in range(ny):
              # check axis handling is self consistent
              assert_array_almost_equal([z_vectest[i], pval_vectest[i]],
                                        stats.mood(x1[i, :], x2[i, :]))
  
      def test_mood_3d(self):
          shape = (10, 5, 6)
          np.random.seed(1234)
          x1 = np.random.randn(*shape)
          x2 = np.random.randn(*shape)
  
          for axis in range(3):
              z_vectest, pval_vectest = stats.mood(x1, x2, axis=axis)
              # Tests that result for 3-D arrays is equal to that for the
              # same calculation on a set of 1-D arrays taken from the
              # 3-D array
              axes_idx = ([1, 2], [0, 2], [0, 1])  # the two axes != axis
              for i in range(shape[axes_idx[axis][0]]):
                  for j in range(shape[axes_idx[axis][1]]):
                      if axis == 0:
                          slice1 = x1[:, i, j]
                          slice2 = x2[:, i, j]
                      elif axis == 1:
                          slice1 = x1[i, :, j]
                          slice2 = x2[i, :, j]
                      else:
                          slice1 = x1[i, j, :]
                          slice2 = x2[i, j, :]
  
                      assert_array_almost_equal([z_vectest[i, j],
                                                 pval_vectest[i, j]],
                                                stats.mood(slice1, slice2))
  
      def test_mood_bad_arg(self):
          # Raise ValueError when the sum of the lengths of the args is less than 3
          assert_raises(ValueError, stats.mood, [1], [])
  
  
  class TestProbplot(TestCase):
  
      def test_basic(self):
          np.random.seed(12345)
          x = stats.norm.rvs(size=20)
          osm, osr = stats.probplot(x, fit=False)
          osm_expected = [-1.8241636, -1.38768012, -1.11829229, -0.91222575,
                          -0.73908135, -0.5857176, -0.44506467, -0.31273668,
                          -0.18568928, -0.06158146, 0.06158146, 0.18568928,
                          0.31273668, 0.44506467, 0.5857176, 0.73908135,
                          0.91222575, 1.11829229, 1.38768012, 1.8241636]
          assert_allclose(osr, np.sort(x))
          assert_allclose(osm, osm_expected)
  
          res, res_fit = stats.probplot(x, fit=True)
          res_fit_expected = [1.05361841, 0.31297795, 0.98741609]
          assert_allclose(res_fit, res_fit_expected)
  
      def test_sparams_keyword(self):
          np.random.seed(123456)
          x = stats.norm.rvs(size=100)
          # Check that None, () and 0 (loc=0, for normal distribution) all work
          # and give the same results
          osm1, osr1 = stats.probplot(x, sparams=None, fit=False)
          osm2, osr2 = stats.probplot(x, sparams=0, fit=False)
          osm3, osr3 = stats.probplot(x, sparams=(), fit=False)
          assert_allclose(osm1, osm2)
          assert_allclose(osm1, osm3)
          assert_allclose(osr1, osr2)
          assert_allclose(osr1, osr3)
          # Check giving (loc, scale) params for normal distribution
          osm, osr = stats.probplot(x, sparams=(), fit=False)
  
      def test_dist_keyword(self):
          np.random.seed(12345)
          x = stats.norm.rvs(size=20)
          osm1, osr1 = stats.probplot(x, fit=False, dist='t', sparams=(3,))
          osm2, osr2 = stats.probplot(x, fit=False, dist=stats.t, sparams=(3,))
          assert_allclose(osm1, osm2)
          assert_allclose(osr1, osr2)
  
          assert_raises(ValueError, stats.probplot, x, dist='wrong-dist-name')
          assert_raises(AttributeError, stats.probplot, x, dist=[])
  
          class custom_dist(object):
              """Some class that looks just enough like a distribution."""
              def ppf(self, q):
                  return stats.norm.ppf(q, loc=2)
  
          osm1, osr1 = stats.probplot(x, sparams=(2,), fit=False)
          osm2, osr2 = stats.probplot(x, dist=custom_dist(), fit=False)
          assert_allclose(osm1, osm2)
          assert_allclose(osr1, osr2)
  
      @dec.skipif(not have_matplotlib)
      def test_plot_kwarg(self):
          np.random.seed(7654321)
          fig = plt.figure()
          fig.add_subplot(111)
          x = stats.t.rvs(3, size=100)
          res1, fitres1 = stats.probplot(x, plot=plt)
          plt.close()
          res2, fitres2 = stats.probplot(x, plot=None)
          res3 = stats.probplot(x, fit=False, plot=plt)
          plt.close()
          res4 = stats.probplot(x, fit=False, plot=None)
          # Check that results are consistent between combinations of `fit` and
          # `plot` keywords.
          assert_(len(res1) == len(res2) == len(res3) == len(res4) == 2)
          assert_allclose(res1, res2)
          assert_allclose(res1, res3)
          assert_allclose(res1, res4)
          assert_allclose(fitres1, fitres2)
  
          # Check that a Matplotlib Axes object is accepted
          fig = plt.figure()
          ax = fig.add_subplot(111)
          stats.probplot(x, fit=False, plot=ax)
          plt.close()
  
      def test_probplot_bad_args(self):
          # Raise ValueError when given an invalid distribution.
          assert_raises(ValueError, stats.probplot, [1], dist="plate_of_shrimp")
  
  
  def test_wilcoxon_bad_arg():
      # Raise ValueError when two args of different lengths are given or
      # zero_method is unknown.
      assert_raises(ValueError, stats.wilcoxon, [1], [1,2])
      assert_raises(ValueError, stats.wilcoxon, [1,2], [1,2], "dummy")
  
  
  def test_mvsdist_bad_arg():
      # Raise ValueError if fewer than two data points are given.
      data = [1]
      assert_raises(ValueError, stats.mvsdist, data)
  
  
  def test_kstat_bad_arg():
      # Raise ValueError if n > 4 or n > 1.
      data = [1]
      n = 10
      assert_raises(ValueError, stats.kstat, data, n=n)
  
  
  def test_kstatvar_bad_arg():
      # Raise ValueError is n is not 1 or 2.
      data = [1]
      n = 10
      assert_raises(ValueError, stats.kstatvar, data, n=n)
  
  
  def test_ppcc_max_bad_arg():
      # Raise ValueError when given an invalid distribution.
      data = [1]
      assert_raises(ValueError, stats.ppcc_max, data, dist="plate_of_shrimp")
  
  
  class TestBoxcox_llf(TestCase):
  
      def test_basic(self):
          np.random.seed(54321)
          x = stats.norm.rvs(size=10000, loc=10)
          lmbda = 1
          llf = stats.boxcox_llf(lmbda, x)
          llf_expected = -x.size / 2. * np.log(np.sum(x.std()**2))
          assert_allclose(llf, llf_expected)
  
      def test_array_like(self):
          np.random.seed(54321)
          x = stats.norm.rvs(size=100, loc=10)
          lmbda = 1
          llf = stats.boxcox_llf(lmbda, x)
          llf2 = stats.boxcox_llf(lmbda, list(x))
          assert_allclose(llf, llf2, rtol=1e-12)
  
      def test_2d_input(self):
          # Note: boxcox_llf() was already working with 2-D input (sort of), so
          # keep it like that.  boxcox() doesn't work with 2-D input though, due
          # to brent() returning a scalar.
          np.random.seed(54321)
          x = stats.norm.rvs(size=100, loc=10)
          lmbda = 1
          llf = stats.boxcox_llf(lmbda, x)
          llf2 = stats.boxcox_llf(lmbda, np.vstack([x, x]).T)
          assert_allclose([llf, llf], llf2, rtol=1e-12)
  
      def test_empty(self):
          assert_(np.isnan(stats.boxcox_llf(1, [])))
  
  
  class TestBoxcox(TestCase):
  
      def test_fixed_lmbda(self):
          np.random.seed(12345)
          x = stats.loggamma.rvs(5, size=50) + 5
          xt = stats.boxcox(x, lmbda=1)
          assert_allclose(xt, x - 1)
          xt = stats.boxcox(x, lmbda=-1)
          assert_allclose(xt, 1 - 1/x)
  
          xt = stats.boxcox(x, lmbda=0)
          assert_allclose(xt, np.log(x))
  
          # Also test that array_like input works
          xt = stats.boxcox(list(x), lmbda=0)
          assert_allclose(xt, np.log(x))
  
      def test_lmbda_None(self):
          np.random.seed(1234567)
          # Start from normal rv's, do inverse transform to check that
          # optimization function gets close to the right answer.
          np.random.seed(1245)
          lmbda = 2.5
          x = stats.norm.rvs(loc=10, size=50000)
          x_inv = (x * lmbda + 1)**(-lmbda)
          xt, maxlog = stats.boxcox(x_inv)
  
          assert_almost_equal(maxlog, -1 / lmbda, decimal=2)
  
      def test_alpha(self):
          np.random.seed(1234)
          x = stats.loggamma.rvs(5, size=50) + 5
  
          # Some regular values for alpha, on a small sample size
          _, _, interval = stats.boxcox(x, alpha=0.75)
          assert_allclose(interval, [4.004485780226041, 5.138756355035744])
          _, _, interval = stats.boxcox(x, alpha=0.05)
          assert_allclose(interval, [1.2138178554857557, 8.209033272375663])
  
          # Try some extreme values, see we don't hit the N=500 limit
          x = stats.loggamma.rvs(7, size=500) + 15
          _, _, interval = stats.boxcox(x, alpha=0.001)
          assert_allclose(interval, [0.3988867, 11.40553131])
          _, _, interval = stats.boxcox(x, alpha=0.999)
          assert_allclose(interval, [5.83316246, 5.83735292])
  
      def test_boxcox_bad_arg(self):
          # Raise ValueError if any data value is negative.
          x = np.array([-1])
          assert_raises(ValueError, stats.boxcox, x)
  
      def test_empty(self):
          assert_(stats.boxcox([]).shape == (0,))
  
  
  class TestBoxcoxNormmax(TestCase):
      def setUp(self):
          np.random.seed(12345)
          self.x = stats.loggamma.rvs(5, size=50) + 5
  
      def test_pearsonr(self):
          maxlog = stats.boxcox_normmax(self.x)
          assert_allclose(maxlog, 1.804465, rtol=1e-6)
  
      def test_mle(self):
          maxlog = stats.boxcox_normmax(self.x, method='mle')
          assert_allclose(maxlog, 1.758101, rtol=1e-6)
  
          # Check that boxcox() uses 'mle'
          _, maxlog_boxcox = stats.boxcox(self.x)
          assert_allclose(maxlog_boxcox, maxlog)
  
      def test_all(self):
          maxlog_all = stats.boxcox_normmax(self.x, method='all')
          assert_allclose(maxlog_all, [1.804465, 1.758101], rtol=1e-6)
  
  
  class TestBoxcoxNormplot(TestCase):
      def setUp(self):
          np.random.seed(7654321)
          self.x = stats.loggamma.rvs(5, size=500) + 5
  
      def test_basic(self):
          N = 5
          lmbdas, ppcc = stats.boxcox_normplot(self.x, -10, 10, N=N)
          ppcc_expected = [0.57783375, 0.83610988, 0.97524311, 0.99756057,
                           0.95843297]
          assert_allclose(lmbdas, np.linspace(-10, 10, num=N))
          assert_allclose(ppcc, ppcc_expected)
  
      @dec.skipif(not have_matplotlib)
      def test_plot_kwarg(self):
          # Check with the matplotlib.pyplot module
          fig = plt.figure()
          fig.add_subplot(111)
          stats.boxcox_normplot(self.x, -20, 20, plot=plt)
          plt.close()
  
          # Check that a Matplotlib Axes object is accepted
          fig.add_subplot(111)
          ax = fig.add_subplot(111)
          stats.boxcox_normplot(self.x, -20, 20, plot=ax)
          plt.close()
  
      def test_invalid_inputs(self):
          # `lb` has to be larger than `la`
          assert_raises(ValueError, stats.boxcox_normplot, self.x, 1, 0)
          # `x` can not contain negative values
          assert_raises(ValueError, stats.boxcox_normplot, [-1, 1], 0, 1)
  
      def test_empty(self):
          assert_(stats.boxcox_normplot([], 0, 1).size == 0)
  
  
  class TestCircFuncs(TestCase):
      def test_circfuncs(self):
          x = np.array([355,5,2,359,10,350])
          M = stats.circmean(x, high=360)
          Mval = 0.167690146
          assert_allclose(M, Mval, rtol=1e-7)
  
          V = stats.circvar(x, high=360)
          Vval = 42.51955609
          assert_allclose(V, Vval, rtol=1e-7)
  
          S = stats.circstd(x, high=360)
          Sval = 6.520702116
          assert_allclose(S, Sval, rtol=1e-7)
  
      def test_circfuncs_small(self):
          x = np.array([20,21,22,18,19,20.5,19.2])
          M1 = x.mean()
          M2 = stats.circmean(x, high=360)
          assert_allclose(M2, M1, rtol=1e-5)
  
          V1 = x.var()
          V2 = stats.circvar(x, high=360)
          assert_allclose(V2, V1, rtol=1e-4)
  
          S1 = x.std()
          S2 = stats.circstd(x, high=360)
          assert_allclose(S2, S1, rtol=1e-4)
  
      def test_circmean_axis(self):
          x = np.array([[355,5,2,359,10,350],
                        [351,7,4,352,9,349],
                        [357,9,8,358,4,356]])
          M1 = stats.circmean(x, high=360)
          M2 = stats.circmean(x.ravel(), high=360)
          assert_allclose(M1, M2, rtol=1e-14)
  
          M1 = stats.circmean(x, high=360, axis=1)
          M2 = [stats.circmean(x[i], high=360) for i in range(x.shape[0])]
          assert_allclose(M1, M2, rtol=1e-14)
  
          M1 = stats.circmean(x, high=360, axis=0)
          M2 = [stats.circmean(x[:,i], high=360) for i in range(x.shape[1])]
          assert_allclose(M1, M2, rtol=1e-14)
  
      def test_circvar_axis(self):
          x = np.array([[355,5,2,359,10,350],
                        [351,7,4,352,9,349],
                    [357,9,8,358,4,356]])
  
          V1 = stats.circvar(x, high=360)
          V2 = stats.circvar(x.ravel(), high=360)
          assert_allclose(V1, V2, rtol=1e-11)
  
          V1 = stats.circvar(x, high=360, axis=1)
          V2 = [stats.circvar(x[i], high=360) for i in range(x.shape[0])]
          assert_allclose(V1, V2, rtol=1e-11)
  
          V1 = stats.circvar(x, high=360, axis=0)
          V2 = [stats.circvar(x[:,i], high=360) for i in range(x.shape[1])]
          assert_allclose(V1, V2, rtol=1e-11)
  
      def test_circstd_axis(self):
          x = np.array([[355,5,2,359,10,350],
                        [351,7,4,352,9,349],
                        [357,9,8,358,4,356]])
  
          S1 = stats.circstd(x, high=360)
          S2 = stats.circstd(x.ravel(), high=360)
          assert_allclose(S1, S2, rtol=1e-11)
  
          S1 = stats.circstd(x, high=360, axis=1)
          S2 = [stats.circstd(x[i], high=360) for i in range(x.shape[0])]
          assert_allclose(S1, S2, rtol=1e-11)
  
          S1 = stats.circstd(x, high=360, axis=0)
          S2 = [stats.circstd(x[:,i], high=360) for i in range(x.shape[1])]
          assert_allclose(S1, S2, rtol=1e-11)
  
      def test_circfuncs_array_like(self):
          x = [355,5,2,359,10,350]
          assert_allclose(stats.circmean(x, high=360), 0.167690146, rtol=1e-7)
          assert_allclose(stats.circvar(x, high=360), 42.51955609, rtol=1e-7)
          assert_allclose(stats.circstd(x, high=360), 6.520702116, rtol=1e-7)
  
      def test_empty(self):
          assert_(np.isnan(stats.circmean([])))
          assert_(np.isnan(stats.circstd([])))
          assert_(np.isnan(stats.circvar([])))
  
  
  def test_accuracy_wilcoxon():
      freq = [1, 4, 16, 15, 8, 4, 5, 1, 2]
      nums = range(-4, 5)
      x = np.concatenate([[u] * v for u, v in zip(nums, freq)])
      y = np.zeros(x.size)
  
      T, p = stats.wilcoxon(x, y, "pratt")
      assert_allclose(T, 423)
      assert_allclose(p, 0.00197547303533107)
  
      T, p = stats.wilcoxon(x, y, "zsplit")
      assert_allclose(T, 441)
      assert_allclose(p, 0.0032145343172473055)
  
      T, p = stats.wilcoxon(x, y, "wilcox")
      assert_allclose(T, 327)
      assert_allclose(p, 0.00641346115861)
  
      # Test the 'correction' option, using values computed in R with:
      # > wilcox.test(x, y, paired=TRUE, exact=FALSE, correct={FALSE,TRUE})
      x = np.array([120, 114, 181, 188, 180, 146, 121, 191, 132, 113, 127, 112])
      y = np.array([133, 143, 119, 189, 112, 199, 198, 113, 115, 121, 142, 187])
      T, p = stats.wilcoxon(x, y, correction=False)
      assert_equal(T, 34)
      assert_allclose(p, 0.6948866, rtol=1e-6)
      T, p = stats.wilcoxon(x, y, correction=True)
      assert_equal(T, 34)
      assert_allclose(p, 0.7240817, rtol=1e-6)
  
  
  def test_wilcoxon_tie():
      # Regression test for gh-2391.
      # Corresponding R code is:
      #   > result = wilcox.test(rep(0.1, 10), exact=FALSE, correct=FALSE)
      #   > result$p.value
      #   [1] 0.001565402
      #   > result = wilcox.test(rep(0.1, 10), exact=FALSE, correct=TRUE)
      #   > result$p.value
      #   [1] 0.001904195
      stat, p = stats.wilcoxon([0.1] * 10)
      expected_p = 0.001565402
      assert_equal(stat, 0)
      assert_allclose(p, expected_p, rtol=1e-6)
  
      stat, p = stats.wilcoxon([0.1] * 10, correction=True)
      expected_p = 0.001904195
      assert_equal(stat, 0)
      assert_allclose(p, expected_p, rtol=1e-6)
  
  
  class TestMedianTest(TestCase):
  
      def test_bad_n_samples(self):
          # median_test requires at least two samples.
          assert_raises(ValueError, stats.median_test, [1, 2, 3])
  
      def test_empty_sample(self):
          # Each sample must contain at least one value.
          assert_raises(ValueError, stats.median_test, [], [1, 2, 3])
  
      def test_empty_when_ties_ignored(self):
          # The grand median is 1, and all values in the first argument are
          # equal to the grand median.  With ties="ignore", those values are
          # ignored, which results in the first sample being (in effect) empty.
          # This should raise a ValueError.
          assert_raises(ValueError, stats.median_test,
                        [1, 1, 1, 1], [2, 0, 1], [2, 0], ties="ignore")
  
      def test_empty_contingency_row(self):
          # The grand median is 1, and with the default ties="below", all the
          # values in the samples are counted as being below the grand median.
          # This would result a row of zeros in the contingency table, which is
          # an error.
          assert_raises(ValueError, stats.median_test, [1, 1, 1], [1, 1, 1])
  
          # With ties="above", all the values are counted as above the
          # grand median.
          assert_raises(ValueError, stats.median_test, [1, 1, 1], [1, 1, 1],
                        ties="above")
  
      def test_bad_ties(self):
          assert_raises(ValueError, stats.median_test, [1, 2, 3], [4, 5], ties="foo")
  
      def test_bad_keyword(self):
          assert_raises(TypeError, stats.median_test, [1, 2, 3], [4, 5], foo="foo")
  
      def test_simple(self):
          x = [1, 2, 3]
          y = [1, 2, 3]
          stat, p, med, tbl = stats.median_test(x, y)
  
          # The median is floating point, but this equality test should be safe.
          assert_equal(med, 2.0)
  
          assert_array_equal(tbl, [[1, 1], [2, 2]])
  
          # The expected values of the contingency table equal the contingency table,
          # so the statistic should be 0 and the p-value should be 1.
          assert_equal(stat, 0)
          assert_equal(p, 1)
  
      def test_ties_options(self):
          # Test the contingency table calculation.
          x = [1, 2, 3, 4]
          y = [5, 6]
          z = [7, 8, 9]
          # grand median is 5.
  
          # Default 'ties' option is "below".
          stat, p, m, tbl = stats.median_test(x, y, z)
          assert_equal(m, 5)
          assert_equal(tbl, [[0, 1, 3], [4, 1, 0]])
  
          stat, p, m, tbl = stats.median_test(x, y, z, ties="ignore")
          assert_equal(m, 5)
          assert_equal(tbl, [[0, 1, 3], [4, 0, 0]])
  
          stat, p, m, tbl = stats.median_test(x, y, z, ties="above")
          assert_equal(m, 5)
          assert_equal(tbl, [[0, 2, 3], [4, 0, 0]])
  
      def test_basic(self):
          # median_test calls chi2_contingency to compute the test statistic
          # and p-value.  Make sure it hasn't screwed up the call...
  
          x = [1, 2, 3, 4, 5]
          y = [2, 4, 6, 8]
  
          stat, p, m, tbl = stats.median_test(x, y)
          assert_equal(m, 4)
          assert_equal(tbl, [[1, 2], [4, 2]])
  
          exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl)
          assert_allclose(stat, exp_stat)
          assert_allclose(p, exp_p)
  
          stat, p, m, tbl = stats.median_test(x, y, lambda_=0)
          assert_equal(m, 4)
          assert_equal(tbl, [[1, 2], [4, 2]])
  
          exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl, lambda_=0)
          assert_allclose(stat, exp_stat)
          assert_allclose(p, exp_p)
  
          stat, p, m, tbl = stats.median_test(x, y, correction=False)
          assert_equal(m, 4)
          assert_equal(tbl, [[1, 2], [4, 2]])
  
          exp_stat, exp_p, dof, e = stats.chi2_contingency(tbl, correction=False)
          assert_allclose(stat, exp_stat)
          assert_allclose(p, exp_p)
  
  
  if __name__ == "__main__":
      run_module_suite()
  