Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-07-06T14:08:35.320Z Has data issue: false hasContentIssue false

A New Distribution of Poisson-Type for the Number of Claims

Published online by Cambridge University Press:  29 August 2014

Michel Denuit*
Affiliation:
Université Libre de Bruxelles, Bruxelles, Belgium
*
Institut de Statistique et de Recherche Opérationnelle, Universite libre de Bruxelles, CP 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgiume-mail:mdenuit@ulb.ac.be
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.

This paper is concerned with two methods to estimate the parameters of the Poisson-Goncharov distribution introduced recently by Lefevre and Picard (1996). These methods are applied to fit, inter alia, the six observed claims distributions, from automobile insurance third party liability portfolios, studied by Gossiaux and Lemaire (1981) and analysed afterwards by several authors.

Type
Articles
Copyright
Copyright © International Actuarial Association 1997

References

REFERENCES

Charalambides, Ch.A. (1990). Abel series distributions with applications to fluctuations of sample functions of stochastic processes. Commun. Statist. A, Theory Methods, 19, 255263.Google Scholar
Consul, P.C. (1989). Generalized Poisson Distributions, Properties and Applications. Marcel Dekker. New York.Google Scholar
Consul, P.C. (1990). A model for distributions of injuries in auto-accidents. MVSV, 161168.Google Scholar
Consul, P.C. and Jain, G.C. (1973). A generalization of the Poisson distribution. Technometrics, 15, 791799.CrossRefGoogle Scholar
Consul, P.C. and Shenton, L.R. (1972). Use of Lagrange expansion for generating discrete generalized probability distributions. SIAM J. Appl. Math., 23, 239248.CrossRefGoogle Scholar
Consul, P.C. and Shoukri, M.M. (1984). Maximum Likelihood estimation for the Generalized Poisson distribution. Comm. Statist. A, Theory Methods, 13, 15331547.Google Scholar
Devroye, L. (1992). The branching process method in Lagrange random variate generation. Commun. Statist., Simula., 21, 114.CrossRefGoogle Scholar
Elvers, E. (1991). A note on the Generalized Poisson distribution. ASTIN Bulletin, 21, 167.Google Scholar
Gerber, H.U. (1990). When does the surplus reach a given target? Insurance: Mathematics and Economics, 9, 115119.Google Scholar
Goovaerts, M.J. and Kaas, R. (1991). Evaluating compound Generalized Poisson distributions recursively. ASTIN Bulletin, 21, 193198.CrossRefGoogle Scholar
Gossiaux, A. M. and Lemaire, J. (1981). Méthodes d'ajustement de distributions de sinistres. MVSV, 8795.Google Scholar
Islam, M.N. and Consul, P.C. (1992). A probabilistic model for automobile claims. MVSV, 8593.Google Scholar
Janardan, K.G., Kerster, H.W. and Schaeffer, D.J. (1979). Biological applications of the Lagrangian Poisson distribution. Bioscience, 29, 599602.CrossRefGoogle Scholar
Kaas, R., Van Heerwaarden, A.E. and Goovaerts, M.J. (1994). Ordering of Actuarial Risks. CAIRE (Education Series). Brussels.Google Scholar
Kestemont, R.-M. and Paris, J. (1985). Sur l'ajustement du nombre de sinistres. MVSV, 157164.Google Scholar
Lefevre, Cl. and Picard, Ph. (1996). On the first crossing of a Poisson process in a lower boundary. In Athens Conference on Applied Probability and Time Series, Vol. I, Applied Probability. Heyde, C.C., Prohorov, Yu V., Pyke, R. and Rachev, S.T. Editors. Lecture Notes in Statistics, 114, pp. 159175. Springer. Berlin.Google Scholar
Lemaire, J. (1991). Negative Binomial or Poisson-Inverse Gaussian? ASTIN Bulletin, 21, 167168.Google Scholar
Lemaire, J. (1995). Bonus-Malus Systems in Automobile Insurance. Klumer Academic Publishers. Boston.CrossRefGoogle Scholar
Oskolkov, V.A. (1988). Abel-Goncharov problem. In Encyclopaedia of Mathematics, Vol. I, pp. 45. Kluwer Academic Publishers. Dordrecht.Google Scholar
Panjer, H.H. (1987). Models of claim frequency. In Advances in the Statistical Sciences, Vol. VI, Actuarial Sciences. Neill, I.B. Mac and Umphrey, G.J. Editors. The University of Western Ontario Series in Philosophy of Science, 39, pp. 115122. Reidel. Dordrecht.Google Scholar
Panjer, H.H. and Willmot, G.E. (1992). Insurance Risk Models. Society of Actuaries. Schaunburg III.Google Scholar
Panjer, H.H. and Wang, S. (1995). Computational aspects of Sundt's generalized class. ASTIN Bulletin, 25, 517.CrossRefGoogle Scholar
Ruohonen, M. (1987). On a model for the claim number process. ASTIN Bulletin, 18, 5768.CrossRefGoogle Scholar
Seal, H.L. (1982). Mixed Poisson – an ideal distribution of claim number?MVSV, 293295.Google Scholar
Shared, M. and Shanthikumar, J.G. (1994). Stochastic Orders and their Applications. Academic Press. New York.Google Scholar
Sharif, A.H. and Panjer, H.H. (1993). A probabilistic model for automobile claims: a comment on the article by M.N. Islam and P.C. Consul. MVSV, 279282.Google Scholar
Sharif, A.H. and Panjer, H.H. (1995). An improved recursion for the compound Generalized Poisson distribution. MVSV, 9398.Google Scholar
Sundt, B. and Jewell, W.S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin, 12, 2739.CrossRefGoogle Scholar
Sundt, B. (1992). On some extension of Panjer's class of counting distributions. ASTIN Bulletin, 22, 6180.CrossRefGoogle Scholar
Ter Berg, P. (1996). A loglinear Lagrangian Poisson model. ASTIN Bulletin, 26, 123129.CrossRefGoogle Scholar
Willmot, G.E. (1987). The Poisson-Inverse Gaussian distribution as an alternative to the Negative Binomial. Scandinavian Actuarial Journal, 113127.CrossRefGoogle Scholar
Willmot, G.E. (1988). Sundt and Jewell's family of discrete distributions. ASTIN Bulletin, 18, 1729.CrossRefGoogle Scholar