statsmodels.genmod.families.family.NegativeBinomial.loglike_obs

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:

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

Return type:

float

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