Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T05:03:04.117Z Has data issue: false hasContentIssue false
  • English
  • Français

An Extension of Panjer's Recursion

Published online by Cambridge University Press:  29 August 2014

Klaus Th. Hess ,
Anett Liewald  and
Klaus D. Schmidt
Klaus Th. Hess*
Affiliation:
Lehrstuhl für Versicherungsmathematik, Technische Universität Dresden
Anett Liewald*
Affiliation:
Lehrstuhl für Versicherungsmathematik, Technische Universität Dresden
Klaus D. Schmidt*
Affiliation:
Lehrstuhl für Versicherungsmathematik, Technische Universität Dresden
*
Lehrstuhl für Versicherungsmathemetik, Technische Universität Dresden, D-01062 Dresden E-mail:schmidt@math.tu-dresden.de
Lehrstuhl für Versicherungsmathemetik, Technische Universität Dresden, D-01062 Dresden E-mail:schmidt@math.tu-dresden.de
Lehrstuhl für Versicherungsmathemetik, Technische Universität Dresden, D-01062 Dresden E-mail:schmidt@math.tu-dresden.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Sundt and Jewell have shown that a nondegenerate claim number distribution Q = {qn}nϵN0 satisfies the recursion

for all n≥0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n ≥ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all n ≥ k with arbitrary k. Our results are of interest in catastrophe excess-of-loss reinsurance.

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 32 , Issue 2 , November 2002 , pp. 283 - 297
Copyright
Copyright © International Actuarial Association 2002

References

DePril, N. (1986) Moments of a class of compound distributions. Scand. Actuar. J., 117–120.CrossRefGoogle Scholar
Hofmann, M. (1955) Über zusammengesetzte Poisson-Prozesse und ihre Anwendungen in der Unfallversicherung. Mitt. Verein. Schweiz. Versicherungsmathematiker 55, 499–575.Google Scholar
Johnson, N.L., Kotz, S. and Kemp, A.W. (1992) Univariate Discrete Distributions. Wiley, New York and Chichester.Google Scholar
Kestemont, R.M. and Paris, J. (1985) Sur l'ajustement du nombre de sinistres. Mitt. Verein. Schweiz. Versicherungsmathematiker 85, 157–164.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (1998) Loss Models. Wiley, New York and Chichester.Google Scholar
Panjer, H.H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bull. 12, 22–26.CrossRefGoogle Scholar
Schmidt, K.D. (1996) Lectures on Risk Theory. Teubner, Stuttgart.CrossRefGoogle Scholar
Schmidt, K.D. (2001) Versicherungsmathematik. Springer, Berlin - Heidelberg - New York.Google Scholar
Sundt, B. (1992) On some extensions of Panjer's class of counting distributions. ASTIN Bull. 22, 61–80.CrossRefGoogle Scholar
Sundt, B. and Jewell, W.S. (1981) Further results of recursive evaluation of compound distributions. ASTIN Bull. 12, 27–39.CrossRefGoogle Scholar
Willmot, G.E. (1988) Sundt and Jewell's family of discrete distributions. ASTIN Bull. 18, 17–29.CrossRefGoogle Scholar
Willmot, G.E. and Lin, X.S. (2001) Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, Berlin - Heidelberg - New York.CrossRefGoogle Scholar