Extensions of Panjer's recursion for mixed compound distributions

TL;DR

This paper extends Panjer's recursion to mixed compound distributions in actuarial science, allowing for dependence and randomness in claim number parameters, with recursive algorithms and numerical illustrations.

Contribution

It introduces a recursive method for mixed compound distributions with dependent claim counts, expanding classical models to more realistic dependent scenarios.

Findings

Recursive algorithm for mixed claim distributions

Extension to exchangeable claim sizes

Numerical examples demonstrating the method

Abstract

In actuarial practice, the usual independence assumptions for the collective risk model are often violated, implying a growing need for considering more general models that incorporate dependence. To this purpose, the present paper studies the mixed counterpart of the classical Panjer family of claim number distributions and their compound version, by allowing the parameters of the distributions to be viewed as random variables. Under the assumptions that the claim size process is conditionally i.i.d. and (conditionally) mutually independent of the claim counts, we provide a recursive algorithm for the computation of the probability mass function of the aggregate claim sizes. The case of a compound Panjer distribution with exchangeable claim sizes is also studied. For the sake of completeness, our results are illustrated by various numerical examples.