Extensions of Panjer’s recursion for mixed compound distributions

In actuarial practice, the usual independence assumptions for the collective risk model are often violated, which implies 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) independent of the claim counts, we provide a recursive formula 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. Numerical examples are also provided to highlight the applicability of this work.