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SOME DISTRIBUTIONAL PROPERTIES OF A CLASS OF COUNTING DISTRIBUTIONS WITH CLAIMS ANALYSIS APPLICATIONS

Published online by Cambridge University Press:  18 June 2013

Gordon E. Willmot  and
Jae-Kyung Woo
Gordon E. Willmot
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada E-Mail: gewillmo@math.uwaterloo.ca
Jae-Kyung Woo*
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong
*
E-Mail: jkwoo@hku.hk
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Abstract

We discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.

Keywords

Mixed Poissoncompound geometricmixed geometricnegative binomiallogarithmic seriescompletely monotoneshifting and truncationDickson–HippGerber–Shiustop-lossequilibrium distributionPanjer recursion
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 43 , Issue 2 , May 2013 , pp. 189 - 212
Copyright
Copyright © ASTIN Bulletin 2013 

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References

Abramowitz, M. and Stegun, I. (1965) Handbook of Mathematical Functions. New York: Dover.Google Scholar
Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics, 29 (3), 333–344.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed. (revised). New York: John Wiley.Google Scholar
Grandell, J. (1997) Mixed Poisson Processes. London: Chapman & Hall.CrossRefGoogle Scholar
Johnson, N., Kemp, A. and Kotz, S. (2005) Univariate Discrete Distributions, 3rd ed. New York: John Wiley.CrossRefGoogle Scholar
Klugman, S., Panjer, H. and Willmot, G. (2008) Loss Models: From Data to Decisions, 3rd ed. New York: John Wiley.CrossRefGoogle Scholar
Lin, X. and Willlmot, G. (1999) Analysis of a defective renewal equation arising in ruin theory. Insurance: Mathematics and Economics, 25, 63–84.Google Scholar
Ord, J. (1972) Families of Frequency Distributions. London: Charles Griffin.Google Scholar
Pavlova, K. and Willmot, G. (2004) The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function. Insurance: Mathematics and Economics, 35, 267–277.Google Scholar
Steutel, F. and Van Harn, K. (2004) Infinite Divisibility of Probability Distributions on the Real Line. New York: Marcel Dekker.Google Scholar
Tröbliger, A. (1961) Mathematische ntersuchungen zur Beitragsruckgewahr in der Kraftfahrversicherung. Blatter der Deutsche Gesellschaft fur Versicherungsmathematik, 5, 327–348.Google Scholar
Willmot, G. (2007) On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics, 41, 17–31.Google Scholar
Willmot, G. (2013) On mixing, compounding, and tail properties of a class of claim number distributions. Scandinavian Actuarial Journal, to appear.CrossRefGoogle Scholar
Willmot, G., Drekic, S. and Cai, J. (2005) Equilibrium compound distributions and stop-loss moments. Scandinavian Actuarial Journal, 2005, 6–24.CrossRefGoogle Scholar
Willmot, G. and Woo, J.-K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 99–118.CrossRefGoogle Scholar