Actuarial statistics with generalized linear mixed models

Over the last decade the use of generalized linear models (GLMs) in actuarial statistics has received a lot of attention, starting from the actuarial illustrations in the standard text by McCullagh and Nelder [McCullagh, P., Nelder, J.A., 1989. Generalized linear models. In: Monographs on Statistics and Applied Probability. Chapman and Hall, New York]. Traditional GLMs however model a sample of independent random variables. Since actuaries very often have repeated measurements or longitudinal data (i.e. repeated measurements over time) at their disposal, this article considers statistical techniques for modelling such data within the framework of GLMs. Use is made of generalized linear mixed models (GLMMs) which model a transformation of the mean as a linear function of both fixed and random effects. The likelihood and Bayesian approaches to GLMMs are explained. The models are illustrated by considering classical credibility models and more general regression models for non-life ratemaking in the context of GLMMs. Details on computation and implementation (in SAS and WinBugs) are provided.