Multivariate lifetime distributions for the exponential dispersion family

We consider a general form of a multivariate lifetime model in which dependence is induced via a common shock component. The univariate marginal distributions come from the well-known and widely applied exponential dispersion family that includes the normal, compound-Poisson, gamma and negative binomial distributions. Any combination of truncation or censoring, either left or right, is considered, for which all moments are derived. This allows for the model to be calibrated to any affine transformation of lifetime data.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

Equilibrium compound distributions and stop-loss moments

A convolution representation is derived for the equilibrium or integrated tail distribution associated with a compound distribution. This result allows for the derivation of reliability properties of compound distributions, as well as an explicit analytic representation for the stop-loss premium, of interest in connection with insurance claims modelling. This result is extended to higher order equilibrium distributions, or equivalently to higher stop-loss moments. Special cases where the counting distribution is mixed Poisson or discrete phase-type are considered in some detail. An approach to handle more general counting distributions is also outlined.