Spatial modelling of claim frequency and claim size in insurance

In this paper models for claim frequency and claim size in non-life
insurance are considered. Both covariates and spatial random e ects
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. Both models for the individual and average claim sizes
of a policyholder are considered. 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 suggested by
Spiegelhalter et al. (2002), the predictive model choice criterion
(Gelfand and Ghosh (1998)) and proper scoring rules (Gneiting and
Raftery (2005)) based on the posterior predictive distribution are
investigated. We give an application to a comprehensive data set
from a German car insurance company. The inclusion of spatial e
ects 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 quantify the significant number of claims e
ects on claim size.