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On the Compound Generalized Poisson Distributions

Published online by Cambridge University Press:  29 August 2014

R.S. Ambagaspitiya  and
N. Balakrishnan
R.S. Ambagaspitiya*
Affiliation:
University of Calgary— McMaster University
N. Balakrishnan*
Affiliation:
University of Calgary— McMaster University
*
Department of Mathematics and Statistis, University of Calgary, Calgary, Alberta, CanadaT2N 1N4.
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, CanadaL8S 4K1.
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Abstract

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Goovaerts and Kaas (1991) present a recursive scheme, involving Panjer's recursion, to compute the compound generalized Poisson distribution (CGPD). In the present paper, we study the CGPD in detail. First, we express the generating functions in terms of Lambert's W function. An integral equation is derived for the pdf of CGPD, when the claim severities are absolutely continuous, from the basic principles. Also we derive the asymptotic formula for CGPD when the distribution of claim severity satisfies certain conditions. Then we present a recursive formula somewhat different and easier to implement than the recursive scheme of Goovaerts and Kaas (1991), when the distribution of claim severity follows an arithmetic distribution, which can be used to evaluate the CGPD. We illustrate the usage of this formula with a numerical example.

Keywords

Compound generalized Poisson distributionsmomentsintegral equationsrecursive equationtail behaviour
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 24 , Issue 2 , November 1994 , pp. 255 - 263
Copyright
Copyright © International Actuarial Association 1994

References

Ambagaspitiya, R.S. and Balakrishnan, N. (1993) Some remarks on Lagragian distributions, Submitted for publication.Google Scholar
Consul, P.C. (1989) Generalized Poisson Distributions : Properties and Applications. Marcel Dekker Inc., New York/Basel.Google Scholar
Consul, P.C. (1990) A model for distributions of injuries in auto-accidents. Itteilungen der Schweiz Vereinigung der Versicherungsmathematiker Heft 1, 161–168.Google Scholar
Consul, P.C. and Jain, G.C. (1973) A generalization of Poisson distribution. Technometrics 15, 791–799.CrossRefGoogle Scholar
Consul, P.C. and Shenton, L.R. (1972) Use of Lagrange expansion for generating discrete generalized probability distributions. SIAM Journal of Applied Mathematics 23, 239–248.CrossRefGoogle Scholar
Consul, P.C. and Shoukri, M.M. (1984) Maximum likelihood estimation for the generalized Poisson distribution. Communications in Statistics- Theory and Methods 10, 977–991.Google Scholar
Consul, P.C. and Shoukri, M.M. (1985) The generalized Poisson distribution when the sample mean is larger than the sample variance. Communications in Statistics — Simulation and Computation 14, 1533–1547.CrossRefGoogle Scholar
Corless, R.M., Gönnet, G. H., Hare, D.E.G. and Jeffrey, D.J. (1994) The Lambert W function. To appear in Advances in Computational Mathematics.Google Scholar
Embrechts, P., Maejima, M. and Teugels, J. (1982) Asymptotic behaviour of compound distributions. ASTIN Bulletin 15, 45–48.CrossRefGoogle Scholar
Goovaerts, M.J. and Kaas, R. (1991) Evaluating compound generalized Poisson distributions recursively. ASTIN Bulletin 21, 193–197.CrossRefGoogle Scholar
Gossiaux, A. and Lemaire, J. (1981) Méthodes d'adjustement de distribution de sinistres. Bulletin of the Association of Swiss Actuaries 81, 87–95.Google Scholar
Panjer, H.H. and Willmot, G.E. (1992) Insurance Risk Models. Society of Actuaries. Seal, H. (1982) Mixed Poisson-an ideal distribution of claim numbers? Bulletin of the Association of Swiss Actuaries 82, 293–295.Google Scholar
Willmot, G.E. (1987) The Poisson-Inverse Gaussian distribution as an alternative to the Negative Binomial. Sandinavian Actuarial Journal, 113–127.CrossRefGoogle Scholar
Willmot, G.E. (1989) Limiting tail behaviour of some discrete compound distributions. Insurance: Mathematics and Economics 8, 175–185.Google Scholar