statsmodels.genmod.families.family.Gaussian.loglike_obs

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

Return type:

float

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

\[ll_i = -1 / 2 \sum_i * var\_weights * ((Y_i - mu_i)^2 / scale + \log(2 * \pi * scale))\]