Fitting generalised linear models and latent class models to data related to claims in motor insurance policies

Abstract

Generalised linear models can accommodate a larger range of model specifications when compared to traditional Normal regression models. Generalised linear models overcome the limitations of Regression models by accommodating any distribution that is a member of the exponential family. These models also allow transformation of the response variable through the link function. Generalised linear models are used to analyze two data sets provided by Middlesea Insurance Company. The first data set comprises information about policy-related variables and individual covariates including the date of birth of the policyholder, the cover subscription and the premium paid annually. The second data set comprises details about all the claims that were made during 2006 including the claim size. The first task is to identify the correct distribution for the claim size and the number of claims filed annually by a claimant. The second task is to investigate the relationships and associations between these car-related variables, by fitting two models to the second data set. The first model is a Log-normal regression model that relates claim size to a number of car-related variables and the second model is a Poisson regression model that relates number of claims filed by a policy holder to these car-related predictors. The two models described above analyze claim size and number of filed claims separately. An appropriate model that describes the aggregate claim amount in a portfolio of insurance contracts during a fixed period combines both claim size and number of claims through a compound distribution. Three compound distributions are presented including the compound Poisson, compound Binomial and compound Negative Binomial distributions. Generalised linear models may yield poor prediction when the explanatory variables explain a small proportion of the variation in the responses. An appropriate model that addresses the heterogeneity of claim sizes is a Latent class model. The purpose of Latent class analysis is to cluster claimants into segments, where the segments are assumed to be homogeneous. A regression model is then derived for each segment using maximum likelihood estimation. Several criteria are employed to determine the optimal number of clusters. The EM algorithm for fitting Latent class models is equivalent to iterative fitting of a weighted generalized linear model with posterior probabilities recalculated at each iteration. The third task makes use of Latent class analysis to derive clusters of claimants such that claimants with similar behaviours are grouped in the same segment.