statsmodels.genmod.families.family.Gaussian.loglike¶

Gaussian.loglike(endog, mu, freq_weights=1.0, scale=1.0)[source]¶

The log-likelihood in terms of the fitted mean response.

Parameters:

endog : array-like

Endogenous response variable

mu : array-like

Fitted mean response variable

freq_weights : array-like

1d array of frequency weights. The default is 1.

scale : float, optional

Scales the loglikelihood function. The default is 1.

Returns:

llf : float

The value of the loglikelihood function evaluated at (endog,mu,freq_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.

$llf = -nobs / 2 * (\log(SSR) + (1 + \log(2 \pi / nobs)))$

where

$SSR = \sum_i (Y_i - g^{-1}(\mu_i))^2$

If the links is not the identity link then the loglikelihood function is defined as

$llf = \sum_i freq\_weights_i * ((Y_i * \mu_i - \mu_i^2 / 2) / scale- Y^2 / (2 * scale) - (1/2) * \log(2 * \pi * scale))$