Select the appropriate model structure for an extended linear model given the behavior of the data set (e.g., appropriate link function and distribution for GLM).

A pricing actuary stares at a column of claim counts: lots of zeros, a handful of ones, the occasional five. Ordinary least squares would happily fit it and predict negative claims. The generalized linear model exists to stop that.

The three components. Every GLM you build on MAS-I is assembled from the same three parts. Get the parts right and the rest is bookkeeping.

• Random component. The response $Y_i$ is independent across $i$ and belongs to a one-parameter exponential family. Its density takes the form below.

$$f(y;\theta ,\varphi )=\exp \left\{\frac{y\theta -b(\theta )}{a(\varphi )}+c(y,\varphi )\right\}$$

The mean is $\mu =b^{\prime }(\theta )$ and the variance is $\text{Var}(Y)=a(\varphi )b^{\prime \prime }(\theta )=a(\varphi )V(\mu )$. The function $V(\mu )$ is the variance function, and it is the fingerprint that distinguishes distributions inside the family.

• Systematic component. A linear predictor ${\eta }_i=x_i^{\top }\beta$ collects the covariates. Linearity is in $\beta$, not in the covariates themselves; you may freely include interactions, polynomials, or splines.

Common mistakes

• Choosing the distribution from the predictor scale rather than the response. A log-transformed response fit with a Normal GLM is not the same as a gamma GLM with log link; Jensen's inequality makes the predictions differ. Pick the distribution from the response.
• Using OLS on counts because "the sample size is large." Even with $n=10,000$, Normal-with-identity predicts negative claim counts and has the wrong variance structure. Predictions and standard errors both mislead.
• Forgetting the exposure offset. Without $\ln (\text{exposure})$, a policy with 0.5 years and one claim looks identical to a policy with 1.0 year and one claim. Coefficients absorb the bias and become uninterpretable.

Bottom line

• A GLM has three pieces: a random component (distribution from the exponential family), a systematic component (linear predictor $\eta =X\beta$), and a link function $g(\mu )=\eta$.
• Pick the distribution from the response support and mean-variance relationship, not the predictor scale. Counts use Poisson or negative binomial, positive severities use gamma or inverse Gaussian, binary outcomes use Bernoulli/binomial, pure premiums use Tweedie.
• Variance functions tie distribution to data: Normal $V(\mu )=1$, Poisson $V(\mu )=\mu$, Gamma $V(\mu )={\mu }^2$, Inverse Gaussian $V(\mu )={\mu }^3$, Tweedie $V(\mu )={\mu }^p$ for $1<p<2$.
• Canonical links (identity for Normal, log for Poisson, logit for binomial, reciprocal for gamma) give sufficient statistics and stable IRLS, but non-canonical links (log on gamma, probit on binary) are routine when interpretability or fit demands it.

Exam shortcut

Match the response to one of five buckets first: binary, count, positive-continuous, symmetric-continuous, or pure-premium-with-zeros. The bucket fixes the distribution before you even read the predictors. Default to the log link for any positive response; only use identity for symmetric continuous and logit/probit for binary.

The full lesson (about 2,395 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C2

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