Mixed Effects Models for Actuarial Applications
Apply mixed effects models with random and fixed effects for actuarial data analysis on Exam MAS-I.
Fixed vs. Random Effects
Mixed effects models include both fixed effects (systematic, reproducible influences like age or gender) and random effects (random variation across groups, such as individual policyholder effects). The model is Y_{ij} = X_{ij}*beta + Z_{ij}*b_i + epsilon_{ij}, where beta are fixed effects, b_i ~ N(0, D) are random effects for group i, and epsilon_{ij} ~ N(0, sigma^2) is residual error. Random effects capture correlation within groups: observations from the same policyholder or territory are correlated through shared random effects.
Actuarial Applications
In insurance, mixed models handle repeated measures (multiple claims per policyholder) and hierarchical data (policies within agencies within regions). The random intercept model allows each group's baseline to vary. Random slope models allow covariate effects to vary by group. Estimation uses restricted maximum likelihood (REML) for variance components and best linear unbiased prediction (BLUP) for random effects. The connection to credibility theory is direct: the BLUP for group means is a credibility-weighted estimate. Exam MAS-I covers mixed model setup, interpretation, and the credibility connection.