We consider claim frequency and average claim size in non-life insurance.Both covariates and spatial random effects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model, we allow for dependencies between claim size and claim frequency. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC).The issue of model comparison is thoroughly addressed. Besides the deviance information criterion (Spiegelhalter et. al 2002) and the predictive model choice criterion (Gelfand and Ghosh 1998), we suggest the use of proper scoring rules (Gneiting and Raftery 2005) based on the posterior predictive distribution for comparing models. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial effects significantly improves the models for both claim frequency and claim size and also leads to more accurate predictions of the total claim sizes. Further we detect significant dependencies between the number of claims and claim size. This is joint work with Susanne Gschloessl.
References: A.E. Gelfand and S.K. Ghosh (1998): Model Choice: A Minimum posterior predictive loss approach, Biometrika, 85 (1), 1-11.
T. Gneiting and A. E. Raftery (2005): Strictly proper scoring rules, prediction and estimation, Technical Report No. 463R, Department of Statistics, University of Washington
D.J. Spiegelhalter, N.G. Best, B.P. Carlin and A. van der Linde (2002): Bayesian measures of model complexity and fit, J. R. Statist. Soc. B, 64 (4), 583-640