Bayesian estimation of ruin probabilities with a heterogeneous and heavy-tailed insurance claim-size distribution

This paper describes a Bayesian approach to make inference for risk reserve processes with an unknown claim-size distribution. A flexible model based on mixtures of Erlang distributions is proposed to approximate the special features frequently observed in insurance claim sizes, such as long tails and heterogeneity. A Bayesian density estimation approach for the claim sizes is implemented using reversible jump Markov chain Monte Carlo methods. An advantage of the considered mixture model is that it belongs to the class of phase-type distributions, and thus explicit evaluations of the ruin probabilities are possible. Furthermore, from a statistical point of view, the parametric structure of the mixtures of the Erlang distribution offers some advantages compared with the whole over-parametrized family of phase-type distributions. Given the observed claim arrivals and claim sizes, we show how to estimate the ruin probabilities, as a function of the initial capital, and predictive intervals that give a measure of the uncertainty in the estimations.