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A Gaussian Exponential Approximation to Some Compound Poisson Distributions

Published online by Cambridge University Press:  17 April 2015

Werner Hürlimann
Werner Hürlimann*
Affiliation:
Winterthur Life and Pensions, Value and Risk Management, Postfach 300 CH-8401, Winterthur, Switzerland
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Abstract

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A three parameter Gaussian exponential approximation to some compound Poisson distributions is considered. It is constructed by specifying the reciprocal of the mean excess function as a linear affine function below some threshold and a positive constant above this threshold. As an analytical approximation to compound Poisson distributions, it is only feasible either for a limited range of the Poisson parameter or for higher coefficients of variation. A semiparametric determination of the unknown threshold parameter is proposed. The analysis of a real-life example from pension fund mathematics displays an improved quality of fit of the new model when compared with other simple good alternative approximations based on the zero gamma, translated gamma and zero translated gamma.

Keywords

Compound Poisson distributionGauss densityexponential densitymean excess functionstop-loss transformanalytical approximationthreshold parameter
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 33 , Issue 1 , May 2003 , pp. 41 - 55
Copyright
Copyright © ASTIN Bulletin 2003

References

Barlow, R.E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Beard, R.E., Pentikäinen, T. and Pesonen, E. (1969/77/84) Risk Theory. The Stochastic Basis of Insurance. Methuen (1969). Chapman and Hall (1977/84).Google Scholar
Benktander, G. (1970) Schadenverteilung nach Grösse in der Nicht-Leben-Versicherung. Bulletin of the Swiss Association of Actuaries, 263-283.Google Scholar
Bhattacharjee, M.C. (1982) The class of mean residual lives and some consequences. SIAM Journal of Algebraic and Discrete Methods 3(1), 56-65.CrossRefGoogle Scholar
Boogaert, P. and De Waegenere, A. (1990) Macro-economic version of a classical formula in risk theory. Insurance: Mathematics and Economics 9, 155-162.Google Scholar
Bryson, M.C. and Sidddiqui, M.M. (1969) Some criteria for aging. Journal of the American Statistical Association 64, 1472-1483.CrossRefGoogle Scholar
Carriere, J. (1992) Limited expected value comparison tests. Statistics and Probability Letters 15, 321-27.Google Scholar
Davis, H.T. and Feldstein, M.L. (1979) The generalized Pareto law as a model for progressively censored survival data. Biometrika 66(2), 299-306.CrossRefGoogle Scholar
Dufresne, F. and Niederhauser, E. (1997) Some analytical approximations of stop-loss premiums. Bulletin of the Swiss Association of Actuaries, 25-47.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, Th. (1997) Modelling Extremal Events for Insurance and Finance. Applications of Mathematics – Stochastic Modelling and Applied Probability, vol. 33. Springer.Google Scholar
Held, R. (1982) Zur rekursiven Berechnung von Stop-Loss Prämien für Pensionskassen. Bulletin of the Swiss Association of Actuaries, 67–89.Google Scholar
Hogg, R. and Klugman, S. (1984) Loss Distributions. John Wiley, New York.CrossRefGoogle Scholar
Hürlimann, W. (1990) Pseudo compound Poisson distributions in risk theory. ASTIN Bulletin 20, 57–79.CrossRefGoogle Scholar
Hürlimann, W. (1994) On stable insurance business models. 25th ASTIN Colloquium, Cannes.Google Scholar
Hürlimann, W. (1996) Improved analytical bounds for some risk quantities. ASTIN Bulletin 26(2), 185-99.CrossRefGoogle Scholar
Hürlimann, W. (1998) Extremal Moment Methods and Stochastic Orders. Application in Actuarial Science. Manuscript (available from the author).Google Scholar
Hürlimann, W. (2000a) Higher-degree stop-loss transforms and stochastic orders (I) theory. Blätter der Deutschen Gesellschaft für Versicherungsmathematik XXIV(3), 449–463.Google Scholar
Hürlimann, W. (2000b) Higher-degree stop-loss transforms and stochastic orders (II) applications. Blätter der Deutschen Gesellschaft für Versicherungsmathematik XXIV(3), 465–476.Google Scholar
Hürlimann, W. (2001) Financial data analysis with two symmetric distributions. First Prize in the Gunnar Benktander ASTIN Award Competition. ASTIN Bulletin 31, 187–211.Google Scholar
Kaas, R., Van Heerwaarden, A.E. and Goovaerts, M.J. (1994) Ordering of Actuarial Risks. CAIRE Education Series 1, Brussels.Google Scholar
Kodlin, D. (1967) A new response time distribution. Biometrics 23, 227–39.CrossRefGoogle ScholarPubMed
Klugman, S., Panjer, H.H. and Willmot, G.E. (1998) Loss Models. From Data to Decisions. John Wiley, New York.Google Scholar
Muth, E.J. (1977) Reliability models with positive memory derived from the mean residual life function. In Tsokas, C.P. and Shimi, I.N. (Editors). The Theory and Applications of Reliability, vol. II, 401–434. Academic Press, New York.Google Scholar
Panjer, H.H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 22–26.CrossRefGoogle Scholar
Panjer, H.H. and Willmot, G.E. (1992) Insurance Risk Models. Society of Actuaries, Schaumburg, Illinois.Google Scholar
Willmot, G.E. (1989) The total claims distribution under inflationary conditions1. Scandinavian Actuarial Journal, 1–12.CrossRefGoogle Scholar