Multivariate Compound Distributions of Type 1

Abstract

AbstractThe main purpose of this chapter is to develop recursions for multivariate compound distributions of Type 1, that is, compound distributions with univariate counting distribution and multivariate severity distribution. We restrict to severity distributions of m×1 vectors of non-negative integers for some fixed integer m. After having given some results on covariances, we consider the case where the counting distribution belongs to the Panjer class, and give some examples. We also extend the multivariate Panjer recursion to counting distributions within the setting of Chap. 5; as a special case, we obtain a recursion for a compound distribution of Type 1 in terms of the severity distribution and the De Pril transform of the counting distribution. From the multivariate Panjer recursion, we deduce recursions for convolutions of multivariate distributions with range bounded on at least one side. We also give a characterisation of a class of infinitely divisible multivariate distributions. Finally, we deduce some recursions for compound distributions with univariate counting distribution and multivariate Bernoulli severity distribution.KeywordsMultivariate DistributionMultivariate CaseCounting DistributionClaim AmountDivisible DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.