text: '''
  The one parameter exponential family distributions used by GLM.
  '''
  # TODO: quasi, quasibinomial, quasipoisson
  # see
  # http://www.biostat.jhsph.edu/~qli/biostatistics_r_doc/library/stats/html/family.html
  # for comparison to R, and McCullagh and Nelder
  
  
  import warnings
  import inspect
  import numpy as np
  from scipy import special
  from . import links as L
  from . import varfuncs as V
  
  FLOAT_EPS = np.finfo(float).eps
  
  
  class Family(object):
      """
      The parent class for one-parameter exponential families.
  
      Parameters
      ----------
      link : a link function instance
          Link is the linear transformation function.
          See the individual families for available links.
      variance : a variance function
          Measures the variance as a function of the mean probabilities.
          See the individual families for the default variance function.
  
      See Also
      --------
      :ref:`links`
      """
      # TODO: change these class attributes, use valid somewhere...
      valid = [-np.inf, np.inf]
      links = []
  
      def _setlink(self, link):
          """
          Helper method to set the link for a family.
  
          Raises a ``ValueError`` exception if the link is not available. Note
          that  the error message might not be that informative because it tells
          you that the link should be in the base class for the link function.
  
          See statsmodels.genmod.generalized_linear_model.GLM for a list of
          appropriate links for each family but note that not all of these are
          currently available.
          """
          # TODO: change the links class attribute in the families to hold
          # meaningful information instead of a list of links instances such as
          # [<statsmodels.family.links.Log object at 0x9a4240c>,
          #  <statsmodels.family.links.Power object at 0x9a423ec>,
          #  <statsmodels.family.links.Power object at 0x9a4236c>]
          # for Poisson...
          self._link = link
          if not isinstance(link, L.Link):
              raise TypeError("The input should be a valid Link object.")
          if hasattr(self, "links"):
              validlink = max([isinstance(link, _) for _ in self.links])
              if not validlink:
                  errmsg = "Invalid link for family, should be in %s. (got %s)"
                  raise ValueError(errmsg % (repr(self.links), link))
  
      def _getlink(self):
          """
          Helper method to get the link for a family.
          """
          return self._link
  
      # link property for each family is a pointer to link instance
      link = property(_getlink, _setlink, doc="Link function for family")
  
      def __init__(self, link, variance):
          if inspect.isclass(link):
              warnmssg = "Calling Family(..) with a link class as argument "
              warnmssg += "is deprecated.\n"
              warnmssg += "Use an instance of a link class instead."
              lvl = 2 if type(self) is Family else 3
              warnings.warn(warnmssg,
                            category=DeprecationWarning, stacklevel=lvl)
              self.link = link()
          else:
              self.link = link
          self.variance = variance
  
      def starting_mu(self, y):
          r"""
          Starting value for mu in the IRLS algorithm.
  
          Parameters
          ----------
          y : array
              The untransformed response variable.
  
          Returns
          -------
          mu_0 : array
              The first guess on the transformed response variable.
  
          Notes
          -----
          .. math::
  
             \mu_0 = (Y + \overline{Y})/2
  
          Only the Binomial family takes a different initial value.
          """
          return (y + y.mean())/2.
  
      def weights(self, mu):
          r"""
          Weights for IRLS steps
  
          Parameters
          ----------
          mu : array_like
              The transformed mean response variable in the exponential family
  
          Returns
          -------
          w : array
              The weights for the IRLS steps
  
          Notes
          -----
          .. math::
  
             w = 1 / (g'(\mu)^2  * Var(\mu))
          """
          return 1. / (self.link.deriv(mu)**2 * self.variance(mu))
  
      def deviance(self, endog, mu, var_weights=1., freq_weights=1., scale=1.):
          r"""
          The deviance function evaluated at (endog, mu, var_weights,
          freq_weights, scale) for the distribution.
  
          Deviance is usually defined as twice the loglikelihood ratio.
  
          Parameters
          ----------
          endog : array_like
              The endogenous response variable
          mu : array_like
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          freq_weights : array_like
              1d array of frequency weights. The default is 1.
          scale : float, optional
              An optional scale argument. The default is 1.
  
          Returns
          -------
          Deviance : array
              The value of deviance function defined below.
  
          Notes
          -----
          Deviance is defined
  
          .. math::
  
             D = 2\sum_i (freq\_weights_i * var\_weights *
             (llf(endog_i, endog_i) - llf(endog_i, \mu_i)))
  
          where y is the endogenous variable. The deviance functions are
          analytically defined for each family.
  
          Internally, we calculate deviance as:
  
          .. math::
             D = \sum_i freq\_weights_i * var\_weights * resid\_dev_i  / scale
          """
          resid_dev = self._resid_dev(endog, mu)
          return np.sum(resid_dev * freq_weights * var_weights / scale)
  
      def resid_dev(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The deviance residuals
  
          Parameters
          ----------
          endog : array_like
              The endogenous response variable
          mu : array_like
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional scale argument. The default is 1.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          The deviance residuals are defined by the contribution D_i of
          observation i to the deviance as
  
          .. math::
             resid\_dev_i = sign(y_i-\mu_i) \sqrt{D_i}
  
          D_i is calculated from the _resid_dev method in each family.
          Distribution-specific documentation of the calculation is available
          there.
          """
          resid_dev = self._resid_dev(endog, mu)
          resid_dev *= var_weights / scale
          return np.sign(endog - mu) * np.sqrt(np.clip(resid_dev, 0., np.inf))
  
      def fitted(self, lin_pred):
          r"""
          Fitted values based on linear predictors lin_pred.
  
          Parameters
          ----------
          lin_pred : array
              Values of the linear predictor of the model.
              :math:`X \cdot \beta` in a classical linear model.
  
          Returns
          -------
          mu : array
              The mean response variables given by the inverse of the link
              function.
          """
          fits = self.link.inverse(lin_pred)
          return fits
  
      def predict(self, mu):
          """
          Linear predictors based on given mu values.
  
          Parameters
          ----------
          mu : array
              The mean response variables
  
          Returns
          -------
          lin_pred : array
              Linear predictors based on the mean response variables.  The value
              of the link function at the given mu.
          """
          return self.link(mu)
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          This is defined for each family. endog and mu are not restricted to
          ``endog`` and ``mu`` respectively.  For instance, you could call
          both ``loglike(endog, endog)`` and ``loglike(endog, mu)`` to get the
          log-likelihood ratio.
          """
          raise NotImplementedError
  
      def loglike(self, endog, mu, var_weights=1., freq_weights=1., scale=1.):
          r"""
          The log-likelihood function in terms of the fitted mean response.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          freq_weights : array_like
              1d array of frequency weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, freq_weights, scale) as defined below.
  
          Notes
          -----
          Where :math:`ll_i` is the by-observation log-likelihood:
  
          .. math::
             ll = \sum(ll_i * freq\_weights_i)
  
          ``ll_i`` is defined for each family. endog and mu are not restricted
          to ``endog`` and ``mu`` respectively.  For instance, you could call
          both ``loglike(endog, endog)`` and ``loglike(endog, mu)`` to get the
          log-likelihood ratio.
          """
          ll_obs = self.loglike_obs(endog, mu, var_weights, scale)
          return np.sum(ll_obs * freq_weights)
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          See Also
          --------
          statsmodels.genmod.families.family.Family : `resid_anscombe` for the
            individual families for more information
  
          Notes
          -----
          Anscombe residuals are defined by
  
          .. math::
             resid\_anscombe_i = \frac{A(y)-A(\mu)}{A'(\mu)\sqrt{Var[\mu]}} *
             \sqrt(var\_weights)
  
          where :math:`A'(y)=v(y)^{-\frac{1}{3}}` and :math:`v(\mu)` is the
          variance function :math:`Var[y]=\frac{\phi}{w}v(mu)`.
          The transformation :math:`A(y)` makes the residuals more normal
          distributed.
          """
          raise NotImplementedError
  
      def _clean(self, x):
          """
          Helper function to trim the data so that it is in (0,inf)
  
          Notes
          -----
          The need for this function was discovered through usage and its
          possible that other families might need a check for validity of the
          domain.
          """
          return np.clip(x, FLOAT_EPS, np.inf)
  
  
  class Poisson(Family):
      """
      Poisson exponential family.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the Poisson family is the log link. Available
          links are log, identity, and sqrt. See statsmodels.families.links for
          more information.
  
      Attributes
      ----------
      Poisson.link : a link instance
          The link function of the Poisson instance.
      Poisson.variance : varfuncs instance
          ``variance`` is an instance of
          statsmodels.genmod.families.varfuncs.mu
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
      """
      links = [L.log, L.identity, L.sqrt]
      variance = V.mu
      valid = [0, np.inf]
      safe_links = [L.Log, ]
  
      def __init__(self, link=None):
          if link is None:
              link = L.log()
          super(Poisson, self).__init__(link=link, variance=Poisson.variance)
  
      def _resid_dev(self, endog, mu):
          r"""
          Poisson deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          .. math::
  
             resid\_dev_i = 2 * (endog_i * \ln(endog_i / \mu_i) -
             (endog_i - \mu_i))
          """
          endog_mu = self._clean(endog / mu)
          resid_dev = endog * np.log(endog_mu) - (endog - mu)
          return 2 * resid_dev
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Poisson distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          .. math::
              ll_i = var\_weights_i / scale * (endog_i * \ln(\mu_i) - \mu_i -
              \ln \Gamma(endog_i + 1))
          """
          return var_weights / scale * (endog * np.log(mu) - mu -
                                        special.gammaln(endog + 1))
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals for the Poisson family defined below
  
          Notes
          -----
          .. math::
  
             resid\_anscombe_i = (3/2) * (endog_i^{2/3} - \mu_i^{2/3}) /
             \mu_i^{1/6} * \sqrt(var\_weights)
          """
          resid = ((3 / 2.) * (endog**(2 / 3.) - mu**(2 / 3.)) /
                   (mu ** (1 / 6.) * scale ** 0.5))
          resid *= np.sqrt(var_weights)
          return resid
  
  
  class Gaussian(Family):
      """
      Gaussian exponential family distribution.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the Gaussian family is the identity link.
          Available links are log, identity, and inverse.
          See statsmodels.genmod.families.links for more information.
  
      Attributes
      ----------
      Gaussian.link : a link instance
          The link function of the Gaussian instance
      Gaussian.variance : varfunc instance
          ``variance`` is an instance of
          statsmodels.genmod.families.varfuncs.constant
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
      """
  
      links = [L.log, L.identity, L.inverse_power]
      variance = V.constant
      safe_links = links
  
      def __init__(self, link=None):
          if link is None:
              link = L.identity()
          super(Gaussian, self).__init__(link=link, variance=Gaussian.variance)
  
      def _resid_dev(self, endog, mu):
          r"""
          Gaussian deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          --------
          .. math::
  
             resid\_dev_i = (endog_i - \mu_i) ** 2
          """
          return (endog - mu) ** 2
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Gaussian distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          If the link is the identity link function then the
          loglikelihood function is the same as the classical OLS model.
  
          .. math::
  
             llf = -nobs / 2 * (\log(SSR) + (1 + \log(2 \pi / nobs)))
  
          where
  
          .. math::
  
             SSR = \sum_i (Y_i - g^{-1}(\mu_i))^2
  
          If the links is not the identity link then the loglikelihood
          function is defined as
  
          .. math::
  
             ll_i = -1 / 2 \sum_i  * var\_weights * ((Y_i - mu_i)^2 / scale +
                                                  \log(2 * \pi * scale))
          """
          ll_obs = -var_weights * (endog - mu) ** 2 / scale
          ll_obs += -np.log(scale / var_weights) - np.log(2 * np.pi)
          ll_obs /= 2
          return ll_obs
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals for the Gaussian family defined below
  
          Notes
          -----
          For the Gaussian distribution, Anscombe residuals are the same as
          deviance residuals.
  
          .. math::
  
             resid\_anscombe_i = (Y_i - \mu_i) / \sqrt{scale} *
             \sqrt(var\_weights)
          """
          resid = (endog - mu) / scale ** 0.5
          resid *= np.sqrt(var_weights)
          return resid
  
  
  class Gamma(Family):
      """
      Gamma exponential family distribution.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the Gamma family is the inverse link.
          Available links are log, identity, and inverse.
          See statsmodels.genmod.families.links for more information.
  
      Attributes
      ----------
      Gamma.link : a link instance
          The link function of the Gamma instance
      Gamma.variance : varfunc instance
          ``variance`` is an instance of
          statsmodels.genmod.family.varfuncs.mu_squared
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
      """
      links = [L.log, L.identity, L.inverse_power]
      variance = V.mu_squared
      safe_links = [L.Log, ]
  
      def __init__(self, link=None):
          if link is None:
              link = L.inverse_power()
          super(Gamma, self).__init__(link=link, variance=Gamma.variance)
  
      def _resid_dev(self, endog, mu):
          r"""
          Gamma deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          .. math::
  
             resid\_dev_i = 2 * ((endog_i - \mu_i) / \mu_i -
             \log(endog_i / \mu_i))
          """
          endog_mu = self._clean(endog / mu)
          resid_dev = -np.log(endog_mu) + (endog - mu) / mu
          return 2 * resid_dev
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Gamma distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          .. math::
  
             ll_i = var\_weights_i / scale * (\ln(var\_weights_i * endog_i /
             (scale * \mu_i)) - (var\_weights_i * endog_i) /
             (scale * \mu_i)) - \ln \Gamma(var\_weights_i / scale) - \ln(\mu_i)
          """
          endog_mu = self._clean(endog / mu)
          weight_scale = var_weights / scale
          ll_obs = weight_scale * np.log(weight_scale * endog_mu)
          ll_obs -= weight_scale * endog_mu
          ll_obs -= special.gammaln(weight_scale) + np.log(endog)
          return ll_obs
  
          # in Stata scale is set to equal 1 for reporting llf
          # in R it's the dispersion, though there is a loss of precision vs.
          # our results due to an assumed difference in implementation
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals for the Gamma family defined below
  
          Notes
          -----
          .. math::
  
             resid\_anscombe_i = 3 * (endog_i^{1/3} - \mu_i^{1/3}) / \mu_i^{1/3}
             / \sqrt{scale} * \sqrt(var\_weights)
          """
          resid = 3 * (endog**(1/3.) - mu**(1/3.)) / mu**(1/3.) / scale ** 0.5
          resid *= np.sqrt(var_weights)
          return resid
  
  
  class Binomial(Family):
      """
      Binomial exponential family distribution.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the Binomial family is the logit link.
          Available links are logit, probit, cauchy, log, and cloglog.
          See statsmodels.genmod.families.links for more information.
  
      Attributes
      ----------
      Binomial.link : a link instance
          The link function of the Binomial instance
      Binomial.variance : varfunc instance
          ``variance`` is an instance of
          statsmodels.genmod.families.varfuncs.binary
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
  
      Notes
      -----
      endog for Binomial can be specified in one of three ways:
      A 1d array of 0 or 1 values, indicating failure or success
      respectively.
      A 2d array, with two columns. The first column represents the
      success count and the second column represents the failure
      count.
      A 1d array of proportions, indicating the proportion of
      successes, with parameter `var_weights` containing the
      number of trials for each row.
      """
  
      links = [L.logit, L.probit, L.cauchy, L.log, L.cloglog, L.identity]
      variance = V.binary  # this is not used below in an effort to include n
  
      # Other safe links, e.g. cloglog and probit are subclasses
      safe_links = [L.Logit, L.CDFLink]
  
      def __init__(self, link=None):  # , n=1.):
          if link is None:
              link = L.logit()
          # TODO: it *should* work for a constant n>1 actually, if freq_weights
          # is equal to n
          self.n = 1
          # overwritten by initialize if needed but always used to initialize
          # variance since endog is assumed/forced to be (0,1)
          super(Binomial, self).__init__(link=link,
                                         variance=V.Binomial(n=self.n))
  
      def starting_mu(self, y):
          r"""
          The starting values for the IRLS algorithm for the Binomial family.
          A good choice for the binomial family is :math:`\mu_0 = (Y_i + 0.5)/2`
          """
          return (y + .5)/2
  
      def initialize(self, endog, freq_weights):
          '''
          Initialize the response variable.
  
          Parameters
          ----------
          endog : array
              Endogenous response variable
          freq_weights : array
              1d array of frequency weights
  
          Returns
          -------
          If `endog` is binary, returns `endog`
  
          If `endog` is a 2d array, then the input is assumed to be in the format
          (successes, failures) and
          successes/(success + failures) is returned.  And n is set to
          successes + failures.
          '''
          # if not np.all(np.asarray(freq_weights) == 1):
          #     self.variance = V.Binomial(n=freq_weights)
          if endog.ndim > 1 and endog.shape[1] > 2:
              raise ValueError('endog has more than 2 columns. The Binomial '
                               'link supports either a single response variable '
                               'or a paired response variable.')
          elif endog.ndim > 1 and endog.shape[1] > 1:
              y = endog[:, 0]
              # overwrite self.freq_weights for deviance below
              self.n = endog.sum(1)
              return y*1./self.n, self.n
          else:
              return endog, np.ones(endog.shape[0])
  
      def _resid_dev(self, endog, mu):
          r"""
          Binomial deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          .. math::
  
             resid\_dev_i = 2 * n * (endog_i * \ln(endog_i /\mu_i) +
             (1 - endog_i) * \ln((1 - endog_i) / (1 - \mu_i)))
          """
          endog_mu = self._clean(endog / mu)
          n_endog_mu = self._clean((1. - endog) / (1. - mu))
          resid_dev = endog * np.log(endog_mu) + (1 - endog) * np.log(n_endog_mu)
          return 2 * self.n * resid_dev
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Binomial distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          If the endogenous variable is binary:
  
          .. math::
  
           ll_i = \sum_i (y_i * \log(\mu_i/(1-\mu_i)) + \log(1-\mu_i)) *
                 var\_weights_i
  
          If the endogenous variable is binomial:
  
          .. math::
  
             ll_i = \sum_i var\_weights_i * (\ln \Gamma(n+1) -
                    \ln \Gamma(y_i + 1) - \ln \Gamma(n_i - y_i +1) + y_i *
                    \log(\mu_i / (n_i - \mu_i)) + n * \log(1 - \mu_i/n_i))
  
          where :math:`y_i = Y_i * n_i` with :math:`Y_i` and :math:`n_i` as
          defined in Binomial initialize.  This simply makes :math:`y_i` the
          original number of successes.
          """
          n = self.n     # Number of trials
          y = endog * n  # Number of successes
  
          # note that mu is still in (0,1), i.e. not converted back
          return (special.gammaln(n + 1) - special.gammaln(y + 1) -
                  special.gammaln(n - y + 1) + y * np.log(mu / (1 - mu)) +
                  n * np.log(1 - mu)) * var_weights
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r'''
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals as defined below.
  
          Notes
          -----
          .. math::
  
              n^{2/3}*(cox\_snell(endog)-cox\_snell(mu)) /
              (mu*(1-mu/n)*scale^3)^{1/6} * \sqrt(var\_weights)
  
          where cox_snell is defined as
          cox_snell(x) = betainc(2/3., 2/3., x)*betainc(2/3.,2/3.)
          where betainc is the incomplete beta function as defined in scipy,
          which uses a regularized version (with the unregularized version, one
          would just have :math:`cox_snell(x) = Betainc(2/3., 2/3., x)`).
  
          The name 'cox_snell' is idiosyncratic and is simply used for
          convenience following the approach suggested in Cox and Snell (1968).
          Further note that
          :math:`cox\_snell(x) = \frac{3}{2}*x^{2/3} *
          hyp2f1(2/3.,1/3.,5/3.,x)`
          where hyp2f1 is the hypergeometric 2f1 function.  The Anscombe
          residuals are sometimes defined in the literature using the
          hyp2f1 formulation.  Both betainc and hyp2f1 can be found in scipy.
  
          References
          ----------
          Anscombe, FJ. (1953) "Contribution to the discussion of H. Hotelling's
              paper." Journal of the Royal Statistical Society B. 15, 229-30.
  
          Cox, DR and Snell, EJ. (1968) "A General Definition of Residuals."
              Journal of the Royal Statistical Society B. 30, 248-75.
          '''
          endog = endog * self.n  # convert back to successes
          mu = mu * self.n  # convert back to successes
  
          def cox_snell(x):
              return special.betainc(2/3., 2/3., x) * special.beta(2/3., 2/3.)
  
          resid = (self.n ** (2/3.) * (cox_snell(endog * 1. / self.n) -
                                       cox_snell(mu * 1. / self.n)) /
                   (mu * (1 - mu * 1. / self.n) * scale ** 3) ** (1 / 6.))
          resid *= np.sqrt(var_weights)
          return resid
  
  
  class InverseGaussian(Family):
      """
      InverseGaussian exponential family.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the inverse Gaussian family is the
          inverse squared link.
          Available links are inverse_squared, inverse, log, and identity.
          See statsmodels.genmod.families.links for more information.
  
      Attributes
      ----------
      InverseGaussian.link : a link instance
          The link function of the inverse Gaussian instance
      InverseGaussian.variance : varfunc instance
          ``variance`` is an instance of
          statsmodels.genmod.families.varfuncs.mu_cubed
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
  
      Notes
      -----
      The inverse Gaussian distribution is sometimes referred to in the
      literature as the Wald distribution.
  
      """
  
      links = [L.inverse_squared, L.inverse_power, L.identity, L.log]
      variance = V.mu_cubed
      safe_links = [L.inverse_squared, L.Log, ]
  
      def __init__(self, link=None):
          if link is None:
              link = L.inverse_squared()
          super(InverseGaussian, self).__init__(
              link=link, variance=InverseGaussian.variance)
  
      def _resid_dev(self, endog, mu):
          r"""
          Inverse Gaussian deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          .. math::
  
             resid\_dev_i = 1 / (endog_i * \mu_i^2) * (endog_i - \mu_i)^2
          """
          return 1. / (endog * mu ** 2) * (endog - mu) ** 2
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Inverse Gaussian distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          .. math::
  
             ll_i = -1/2 * (var\_weights_i * (endog_i - \mu_i)^2 /
             (scale * endog_i * \mu_i^2) + \ln(scale * \endog_i^3 /
             var\_weights_i) - \ln(2 * \pi))
          """
          ll_obs = -var_weights * (endog - mu) ** 2 / (scale * endog * mu ** 2)
          ll_obs += -np.log(scale * endog ** 3 / var_weights) - np.log(2 * np.pi)
          ll_obs /= 2
          return ll_obs
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals for the inverse Gaussian distribution  as
              defined below
  
          Notes
          -----
          .. math::
  
             resid\_anscombe_i = \log(Y_i / \mu_i) / \sqrt{\mu_i * scale} *
             \sqrt(var\_weights)
          """
          resid = np.log(endog / mu) / np.sqrt(mu * scale)
          resid *= np.sqrt(var_weights)
          return resid
  
  
  class NegativeBinomial(Family):
      r"""
      Negative Binomial exponential family.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the negative binomial family is the log link.
          Available links are log, cloglog, identity, nbinom and power.
          See statsmodels.genmod.families.links for more information.
      alpha : float, optional
          The ancillary parameter for the negative binomial distribution.
          For now ``alpha`` is assumed to be nonstochastic.  The default value
          is 1.  Permissible values are usually assumed to be between .01 and 2.
  
      Attributes
      ----------
      NegativeBinomial.link : a link instance
          The link function of the negative binomial instance
      NegativeBinomial.variance : varfunc instance
          ``variance`` is an instance of
          statsmodels.genmod.families.varfuncs.nbinom
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
  
      Notes
      -----
      Power link functions are not yet supported.
  
      Parameterization for :math:`y=0, 1, 2, \ldots` is
  
      .. math::
  
         f(y) = \frac{\Gamma(y+\frac{1}{\alpha})}{y!\Gamma(\frac{1}{\alpha})}
                \left(\frac{1}{1+\alpha\mu}\right)^{\frac{1}{\alpha}}
                \left(\frac{\alpha\mu}{1+\alpha\mu}\right)^y
  
      with :math:`E[Y]=\mu\,` and :math:`Var[Y]=\mu+\alpha\mu^2`.
      """
      links = [L.log, L.cloglog, L.identity, L.nbinom, L.Power]
      # TODO: add the ability to use the power links with an if test
      # similar to below
      variance = V.nbinom
      safe_links = [L.Log, ]
  
      def __init__(self, link=None, alpha=1.):
          self.alpha = 1. * alpha  # make it at least float
          if link is None:
              link = L.log()
          super(NegativeBinomial, self).__init__(
              link=link, variance=V.NegativeBinomial(alpha=self.alpha))
  
      def _resid_dev(self, endog, mu):
          r"""
          Negative Binomial deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          .. math::
  
              resid_dev_i = 2 * (endog_i * \ln(endog_i /
              \mu_i) - (endog_i + 1 / \alpha) * \ln((endog_i + 1 / \alpha) /
              (\mu_i + 1 / \alpha)))
          """
          endog_mu = self._clean(endog / mu)
          endog_alpha = endog + 1 / self.alpha
          mu_alpha = mu + 1 / self.alpha
          resid_dev = endog * np.log(endog_mu)
          resid_dev -= endog_alpha * np.log(endog_alpha / mu_alpha)
          return 2 * resid_dev
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Negative Binomial distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          Defined as:
  
          .. math::
  
             llf = \sum_i var\_weights_i / scale * (Y_i * \log{(\alpha * \mu_i /
                   (1 + \alpha * \mu_i))} - \log{(1 + \alpha * \mu_i)}/
                   \alpha + Constant)
  
          where :math:`Constant` is defined as:
  
          .. math::
  
             Constant = \ln \Gamma{(Y_i + 1/ \alpha )} - \ln \Gamma(Y_i + 1) -
                        \ln \Gamma{(1/ \alpha )}
  
          constant = (special.gammaln(endog + 1 / self.alpha) -
                      special.gammaln(endog+1)-special.gammaln(1/self.alpha))
          return (endog * np.log(self.alpha * mu / (1 + self.alpha * mu)) -
                  np.log(1 + self.alpha * mu) / self.alpha +
                  constant) * var_weights / scale
          """
          ll_obs = endog * np.log(self.alpha * mu)
          ll_obs -= (endog + 1 / self.alpha) * np.log(1 + self.alpha * mu)
          ll_obs += special.gammaln(endog + 1 / self.alpha)
          ll_obs -= special.gammaln(1 / self.alpha)
          ll_obs -= special.gammaln(endog + 1)
          return var_weights / scale * ll_obs
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals as defined below.
  
          Notes
          -----
          Anscombe residuals for Negative Binomial are the same as for Binomial
          upon setting :math:`n=-\frac{1}{\alpha}`. Due to the negative value of
          :math:`-\alpha*Y` the representation with the hypergeometric function
          :math:`H2F1(x) =  hyp2f1(2/3.,1/3.,5/3.,x)` is advantageous
  
          .. math::
  
              resid\_anscombe_i = \frac{3}{2} *
              (Y_i^(2/3)*H2F1(-\alpha*Y_i) - \mu_i^(2/3)*H2F1(-\alpha*\mu_i))
              / (\mu_i * (1+\alpha*\mu_i) * scale^3)^(1/6) * \sqrt(var\_weights)
  
          Note that for the (unregularized) Beta function, one has
          :math:`Beta(z,a,b) = z^a/a * H2F1(a,1-b,a+1,z)`
          """
          def hyp2f1(x):
              return special.hyp2f1(2 / 3., 1 / 3., 5 / 3., x)
  
          resid = (3 / 2. * (endog ** (2 / 3.) * hyp2f1(-self.alpha * endog) -
                             mu ** (2 / 3.) * hyp2f1(-self.alpha * mu)) /
                   (mu * (1 + self.alpha * mu) *
                   scale ** 3) ** (1 / 6.))
          resid *= np.sqrt(var_weights)
          return resid
  
  
  class Tweedie(Family):
      """
      Tweedie family.
  
      Parameters
      ----------
      link : a link instance, optional
          The default link for the Tweedie family is the log link.
          Available links are log and Power.
          See statsmodels.genmod.families.links for more information.
      var_power : float, optional
          The variance power. The default is 1.
      eql : bool
          If True, the Extended Quasi-Likelihood is used, else the
          likelihood is used (however the latter is not implemented).
          If eql is True, var_power must be between 1 and 2.
  
      Attributes
      ----------
      Tweedie.link : a link instance
          The link function of the Tweedie instance
      Tweedie.variance : varfunc instance
          ``variance`` is an instance of
          statsmodels.genmod.families.varfuncs.Power
      Tweedie.var_power : float
          The power of the variance function.
  
      See Also
      --------
      statsmodels.genmod.families.family.Family
      :ref:`links`
  
      Notes
      -----
      Loglikelihood function not implemented because of the complexity of
      calculating an infinite series of summations. The variance power can be
      estimated using the ``estimate_tweedie_power`` function that is part of the
      statsmodels.genmod.generalized_linear_model.GLM class.
      """
      links = [L.log, L.Power]
      variance = V.Power(power=1.5)
      safe_links = [L.log, L.Power]
  
      def __init__(self, link=None, var_power=1., eql=False):
          self.var_power = var_power
          self.eql = eql
          if eql and (var_power < 1 or var_power > 2):
              raise ValueError("Tweedie: if EQL=True then var_power must fall "
                               "between 1 and 2")
          if link is None:
              link = L.log()
          super(Tweedie, self).__init__(
              link=link, variance=V.Power(power=var_power * 1.))
  
      def _resid_dev(self, endog, mu):
          r"""
          Tweedie deviance residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable.
          mu : array
              The inverse of the link function at the linear predicted values.
  
          Returns
          -------
          resid_dev : float
              Deviance residuals as defined below.
  
          Notes
          -----
          When :math:`p = 1`,
  
          .. math::
  
              dev_i = \mu_i
  
          when :math:`endog_i = 0` and
  
          .. math::
  
              dev_i = endog_i * \log(endog_i / \mu_i) + (\mu_i - endog_i)
  
          otherwise.
  
          When :math:`p = 2`,
  
          .. math::
  
              dev_i =  (endog_i - \mu_i) / \mu_i - \log(endog_i / \mu_i)
  
          For all other p,
  
          .. math::
  
              dev_i = endog_i^{2 - p} / ((1 - p) * (2 - p)) -
                      endog_i * \mu_i^{1 - p} / (1 - p) + \mu_i^{2 - p} /
                      (2 - p)
  
          The deviance residual is then
  
          .. math::
  
              resid\_dev_i = 2 * dev_i
          """
          p = self.var_power
          if p == 1:
              dev = np.where(endog == 0,
                             mu,
                             endog * np.log(endog / mu) + (mu - endog))
          elif p == 2:
              endog1 = self._clean(endog)
              dev = ((endog - mu) / mu) - np.log(endog1 / mu)
          else:
              dev = (endog ** (2 - p) / ((1 - p) * (2 - p)) -
                     endog * mu ** (1-p) / (1 - p) + mu ** (2 - p) / (2 - p))
          return 2 * dev
  
      def loglike_obs(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The log-likelihood function for each observation in terms of the fitted
          mean response for the Tweedie distribution.
  
          Parameters
          ----------
          endog : array
              Usually the endogenous response variable.
          mu : array
              Usually but not always the fitted mean response variable.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float
              The scale parameter. The default is 1.
  
          Returns
          -------
          ll_i : float
              The value of the loglikelihood evaluated at
              (endog, mu, var_weights, scale) as defined below.
  
          Notes
          -----
          If eql is True, the Extended Quasi-Likelihood is used.  At present,
          this method returns NaN if eql is False.  When the actual likelihood
          is implemented, it will be accessible by setting eql to False.
  
          References
          ----------
          JA Nelder, D Pregibon (1987).  An extended quasi-likelihood function.
          Biometrika 74:2, pp 221-232.  https://www.jstor.org/stable/2336136
          """
          if not self.eql:
              # We have not yet implemented the actual likelihood
              return np.nan
  
          # Equations 9-10 or Nelder and Pregibon
          p = self.var_power
          llf = np.log(2 * np.pi * scale) + p * np.log(mu) - np.log(var_weights)
          llf /= -2
  
          if p == 1:
              u = endog * np.log(endog / mu) - (endog - mu)
              u *= var_weights / scale
          elif p == 2:
              yr = endog / mu
              u = yr - np.log(yr) - 1
              u *= var_weights / scale
          else:
              u = (endog ** (2 - p)
                   - (2 - p) * endog * mu ** (1 - p)
                   + (1 - p) * mu ** (2 - p))
              u *= var_weights / (scale * (1 - p) * (2 - p))
          llf -= u
  
          return llf
  
      def resid_anscombe(self, endog, mu, var_weights=1., scale=1.):
          r"""
          The Anscombe residuals
  
          Parameters
          ----------
          endog : array
              The endogenous response variable
          mu : array
              The inverse of the link function at the linear predicted values.
          var_weights : array_like
              1d array of variance (analytic) weights. The default is 1.
          scale : float, optional
              An optional argument to divide the residuals by sqrt(scale).
              The default is 1.
  
          Returns
          -------
          resid_anscombe : array
              The Anscombe residuals as defined below.
  
          Notes
          -----
          When :math:`p = 3`, then
  
          .. math::
  
              resid\_anscombe_i = \log(endog_i / \mu_i) / \sqrt{\mu_i * scale} *
              \sqrt(var\_weights)
  
          Otherwise,
  
          .. math::
  
              c = (3 - p) / 3
  
          .. math::
  
              resid\_anscombe_i = (1 / c) * (endog_i^c - \mu_i^c) / \mu_i^{p / 6}
              / \sqrt{scale} * \sqrt(var\_weights)
          """
          if self.var_power == 3:
              resid = np.log(endog / mu) / np.sqrt(mu * scale)
          else:
              c = (3. - self.var_power) / 3.
              resid = ((1. / c) * (endog ** c - mu ** c) /
                       mu ** (self.var_power / 6.)) / scale ** 0.5
          resid *= np.sqrt(var_weights)
          return resid
  