Smoothed NPML 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.