Select the appropriate model for an extended linear model.

A pricing actuary has a portfolio with skewed claim counts, heteroskedastic severities, and a dozen candidate rating variables. Ordinary least squares will mislead. Model selection inside the extended linear family (GLMs, penalized regression, GAMs) is what MAS-I tests, and the choice rests on the response distribution, the link, and a defensible comparison metric.

The extended linear family in one frame. A generalized linear model (GLM) has three pieces: a random component (response distribution in the exponential family), a systematic component (the linear predictor $\eta = X\beta$), and a link function $g$ connecting them through $g(\mu) = \eta$. Selecting a model means choosing all three plus the variables that enter $X$.

KEY: Distribution, link, and predictors are three separate decisions. A wrong link can ruin a correctly specified distribution, and vice versa.

Choosing the response distribution. The variance structure of the data is the single best guide.

Why these variance functions matter. The mean-variance relationship implied by the distribution determines weighting in the iteratively reweighted least squares fit.

Common mistakes

• Picking the canonical link by reflex. Gamma's canonical link is the inverse, but log is standard practice because it enforces positivity and gives multiplicative interpretation. Choose by interpretability and constraint, not by what is "canonical".
• Comparing AIC across different response distributions. AIC values are only comparable when the log-likelihood is on the same scale. A Poisson AIC and a gamma AIC measure different things; do not put them side by side.
• Forgetting to standardize before penalizing. Without standardization, lasso and ridge over-penalize coefficients on small-scale predictors and under-penalize large-scale ones. Always center and scale continuous covariates first.

Bottom line

• Match the distribution to the response: counts use Poisson or negative binomial, positive skewed severities use gamma or inverse Gaussian, binary outcomes use binomial, pure-premium responses use Tweedie.
• Match the variance function to the data: Poisson $V=\mu$, gamma $V=\mu^2$, inverse Gaussian $V=\mu^3$.
• Match the link to the scale: log link for multiplicative rating, logit for probabilities, identity only when an additive structure is defensible.
• Tweedie with $1 < p < 2$ handles pure-premium responses that carry a point mass at zero plus continuous positive losses.

Exam shortcut

When the prompt mentions "claim counts" with no further hint, default to Poisson with log link; if it adds "over-dispersed" or gives $\text{Var}/\text{mean}$ above 1.2, switch to negative binomial. When the prompt mentions "right-skewed positive" or "severity", default to gamma with log link.

The full lesson (about 3,212 words, 21 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C1

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