statsmodels.genmod.families.family.NegativeBinomial.loglike_obs

method

NegativeBinomial.loglike_obs(endog, mu, var_weights=1.0, scale=1.0)

The log-likelihood function for each observation in terms of the fitted mean response for the Negative Binomial distribution.

Parameters

endogarray

Usually the endogenous response variable.

muarray

Usually but not always the fitted mean response variable.

var_weightsarray-like

1d array of variance (analytic) weights. The default is 1.

scalefloat

The scale parameter. The default is 1.

Returns

ll_ifloat

The value of the loglikelihood evaluated at (endog, mu, var_weights, scale) as defined below.

Notes

Defined as:

\[llf = \sum_i var\_weights_i / scale * (Y_i * \log{(\alpha * \mu_i / (1 + \alpha * \mu_i))} - \log{(1 + \alpha * \mu_i)}/ \alpha + Constant)\]

where \(Constant\) is defined as:

\[Constant = \ln \Gamma{(Y_i + 1/ \alpha )} - \ln \Gamma(Y_i + 1) - \ln \Gamma{(1/ \alpha )}\]

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