Generalized Linear Models in Depth for MAS-I

GLM Components

A generalized linear model has three components: (1) a random component specifying the response distribution from the exponential family (Normal, Poisson, Binomial, Gamma, Inverse Gaussian), (2) a systematic component eta = X*beta (linear predictor), and (3) a link function g connecting the mean to the linear predictor: g(mu) = eta. The canonical link for each distribution simplifies estimation: identity for Normal, log for Poisson, logit for Binomial, and reciprocal for Gamma. The variance function V(mu) characterizes the mean-variance relationship.

Estimation and Inference

GLM parameters are estimated by maximum likelihood using iteratively reweighted least squares (IRLS). The deviance D = 2*(ln L_saturated minus ln L_model) measures goodness of fit and is analogous to the residual sum of squares in linear regression. Comparing nested models uses the deviance difference, which is approximately chi-squared. Pearson and deviance residuals diagnose model fit. AIC and BIC guide model selection. Exam MAS-I tests GLM setup for insurance applications (claim frequency as Poisson, severity as Gamma), parameter interpretation, and model comparison.