Designing an Optimal Bonus--Malus System Using the Number of Reported Claims, Steady-State Distribution, and Mixture Claim Size Distribution

This article, in a first step, considers two Bayes estimators for the relativity premium of a given Bonus--Malus system. It then develops a linear relativity premium that closes, in the sense of weighted mean square error loss, to such Bayes estimators. In a second step, it supposes that the claim size distribution for a given Bonus--Malus system can be formulated as a finite mixture distribution. It then evaluates the base premium under a Bayesian framework for such a finite mixture distribution. The Loimaranta efficiency of such a linear relativity premium, for several Bonus--Malus systems, has been compared with two Bayes and ordinary linear relativity premiums.