On some Extensions of Panjer's Class of Counting Distributions

Abstract

In this paper we discuss some properties of counting distributions whose discrete density {p n }% =0 satisfies a recursion in the form k ( « = 1,2,...) with p n = 0 for n < 0 and present an algorithm for recursive evaluation of corresponding compound distributions.

On Multivariate Panjer Recursions

“…These recursions reduce to respectively (3.3) and (3.8) when r -\, and to the recursion of Sundt (1992) when m = 1. The recursions of the present subsection are further analysed in Sundt (1998 4A.…”

“…When r = 1, (3.9) reduces to (1.2). By modifying the proof of Theorem 9 in Sundt (1992) analogous to the way we modified the proof of Theorem 10.6 in Sundt (1993) for the proof of Theorem 1, we obtain " u) t and analogous to (3.8) we obtain…”

The Concepts of Pseudo Compound Poisson and Partition Representations in Discrete Probability

“…Otherwise, one says that has a pseudo compound Poisson distribution. In the terminology of Sundt [8], it belongs to the class ∞ [0, ] with ( ) = ( ) (see also [9]). The theoretical and practical usefulness of pseudo compound Poisson distributions have been demonstrated by the author in numerous publications.…”

A generalization of Panjer’s recursion and numerically stable risk aggregation

“…These algorithms can, for example, be Panjer's recursion in its usual form for distributions of the Panjer(a, b, k) class where it is stable. Furthermore, the algorithms given by Sundt [28] or by Wang and Sobrero [31] for the corresponding class can be applied, whenever they can be shown to be numerically stable. Further examples are, of course, algorithms using our results, like Algorithm 5.3 for ExtNegBin(↵, k, p), Algorithm 5.6 for ExtLog(k, q), Algorithms 5.12 and 5.18 for the extended tempered stable distributions given by (5.21), and extensions by convolution outlined in Section 5.4, in particular Example 5.24.…”

“…More general relations than (1.2) and corresponding recursion schemes have been considered in articles by Sundt [28], Hesselager [12], and Wang and Sobrero [31]. Panjer and Wang [21] show that, for non-degenerate severity distributions, the numerical stability of Panjer's recursion with claim number distribution in the Panjer(a, b, k) class only depends on the values of a and b.…”