In this talk models for claim frequency and average claim size in non-life insurance are considered. 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 going back to Lundberg (1903), 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 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. Both spatial and number of claims effects are interpreted and quantified from an actuarial point of view. Model extension including over dispersion models such as zero inflated and generalized Poisson distributions (Consul 1973) will be also discussed.
This work is joint with Susanne Gschloessl. Background material can be found in:
GschlÃ¶ÃŸl, S. and Czado, C. (2006) Spatial modelling of claim frequency and claim size in non-life insurance. Submitted for publication.
GschlÃ¶ÃŸl, S. and Czado, C. (2005) Modelling count data with over dispersion and spatial effects. Submitted for publication.