Smoothed nonparametric maximum likelihood estimation of the risk distribution underlying bonus-malus systems

Mixed Poisson distributions are widely used for modeling
claim counts when the portfolio is thought to be
heterogeneous. The risk (or mixing) distribution then
represents a measure of this heterogeneity. The aim of
this paper is to use a variant of the Patilea and Rolin
[15] smoothed version of the Simar [20] Non-Parametric
Maximum Likelihood Estimator of the risk distribution
in the mixed Poisson model. Empirical results based on
two data sets from automobile third-party liability insurance
demonstrate the relevance of this approach. The
design of merit-rating schemes is discussed in the second
part of the paper.