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Mixed Compound Poisson Distributions*

Published online by Cambridge University Press:  29 August 2014

Gord Willmot
Gord Willmot*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo
*
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1
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Abstract

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The distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).

Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.

It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

Keywords

Mixed Poissoncompound PoissonrecursionsVolterra integral equationsinverse GaussianPoisson mixturesNeyman contagious distributionspower mixturesqueuesinfinite divisibilitycompound distributions
Type
Astin Competition 1985: Prize-Winning Papers and Other Selected Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 16 , Issue S1 , April 1986 , pp. S59 - S79
Copyright
Copyright © International Actuarial Association 1986

Footnotes

*

This research was supported by the Natural Sciences and Engineering Research Council of Canada. The author is indebted to an anonymous referee for numerous suggestions.

References

Atkinson, A. C. and Yeh, Lam (1982). Inference for Sichel's Compound Poisson Distribution. Journal of the American Statistical Association 77, 153–158.CrossRefGoogle Scholar
Baker, C. T. (1977) The Numerical Treatment of Integral Equations. Clarendon Press: Oxford.Google Scholar
Beall, G. and Rescia, R. (1953) A Generalization of Neyman's Contagious Distributions. Biometrics 9, 354–386.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. Springer-Verlag: New York.Google Scholar
Bühlmann, H. and Buzzi, R. (1970) On a Transformation of the Weighted Compound Poisson Process. Astin Bulletin 6, 42–46.Google Scholar
Douglas, J. B. (1980) Analysis with Standard Contagious Distributions. International Co-operative Publishing House: Fairland, Maryland.Google Scholar
Embrechts, P. (1983) A Property of the Generalized Inverse Gaussian Distribution with Some Applications. Journal of Applied Probability 20, 537–544.CrossRefGoogle Scholar
Engen, S. (1974) On Species Frequency Models. Biometrika 61, 263–270.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications, vol. 1 (3rd ed.). John Wiley: New York.Google Scholar
Haight, F. (1967) Handbook of the Poisson Distribution. John Wiley: New York.Google Scholar
Holgate, P. (1970) The Modality of Some Compound Poisson Distributions. Biometrika 57, 666–667.CrossRefGoogle Scholar
Holla, M. S. (1967) On A Poisson-Inverse Gaussian Distribution. Metrika 11, 115–121.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1969). Distributions in Statistics: Discrete Distributions. John Wiley: New York.Google Scholar
Johnson, N. L. and Kotz, S. (1970a) Distributions in Statistics: Continuous Univariate Distributions, 1. John Wiley: New York.Google Scholar
Johnson, N. L. and Kotz, S. (19706) Distributions in Statistics: Continuous Univariate Distributions 2. John Wiley: New York.Google Scholar
Jørgensen, B. (1982) Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics 9. Springer-Verlag: New York.CrossRefGoogle Scholar
Kleinrock, L. (1975) Queueing Systems, Volume 1, Theory. John Wiley, New York.Google Scholar
Lundberg, O. (1940) On Random Processes and Their Application to Sickness and Accident Statistics. Almquist and Wiksell: Uppsala.Google Scholar
Maceda, E. C. (1948) On the Compound and Generalized Poisson Distributions. Annals of Mathematical Statistics 19, 414–416.CrossRefGoogle Scholar
McFadden, J. A. (1965) the Mixed Poisson Process. Sankhya A 27, 83–92.Google Scholar
Ord, J. (1972) Families of Frequency Distributions. Charles Griffin: London.Google Scholar
Panjer, H. H. (1981) Recursive Evaluation of A Family of Compound Distributions. Astin Bulletin 12, 22–26.CrossRefGoogle Scholar
Sichel, H. S. (1971) On A Family of Discrete Distributions Particularly Suited to Represent Long Tailed Frequency Data. Proceedings of the Third Symposium on Mathematical Statistics, ed. Loubscher, N. F.. Pretoria: CSIR.Google Scholar
Ströter, B. (1984) The Numerical Evaluation of the Aggregate Claim Density Function Via Integral Equations. Blätter der Deutschen Gesellschaft für Versicherungs-mathematik 17, 1–14.Google Scholar
Sundt, B. and Jewell, W. (1981) Further Results on Recursive Evaluation of Compound Distributions. Astin Bulletin 12, 27–39.CrossRefGoogle Scholar
Van Harn, K. (1978) Classifying Infinitely Divisible Distributions By Functional Equations. Math. Centre Tracts 103: Math. Centre, Amsterdam.Google Scholar
Willmot, G. E. and Panjer, H. H. (1985) Difference Equation Approaches in Evaluation of Compound Distributions. To appear.Google Scholar