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A New Class of Bayesian Estimators in Paretian Excess-of-Loss Reinsurance

Published online by Cambridge University Press:  29 August 2014

R.-D. Reiss  and
M. Thomas
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Abstract

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For estimating the shape parameter of Paretian excess claims, certain Bayesian estimators, which are closely related to the Hill estimator, have been suggested in the insurance literature. It turns out that these estimators may have a poor performance – just as the Hill estimator – if a certain location parameter is unequal to zero in the Paretian modeling. In an alternative formulation this means that a scale parameter is unequal to 1. Thus, it suggests itself to add the scale parameter in the modeling and to deal with Bayesian estimators of the shape and scale parameters in a full Paretian model. These estimators will be applied to fire and motor reinsurance data. The performance of these estimators will be illustrated by means of Monte Carlo simulations.

Keywords

Excess claimsFire and motor reinsuranceFull Pareto modelGeneralized Pareto modelBayesian estimatorsHill estimatorMaximum likelihood estimator
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 29 , Issue 2 , November 1999 , pp. 339 - 349
Copyright
Copyright © International Actuarial Association 1999

References

Arnold, B.C. and Press, S.J. (1989). Bayesian estimation and prediction for Pareto data. JASA 84, 1079–1084.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, New York.CrossRefGoogle Scholar
Falk, M., Hüsler, J. and Reiss, R.-D. (1993). Laws of Small Numbers: Extremes and Rare Events. DMV Seminar 23, Birkhäuser, Basel.Google Scholar
Hartigan, J.A. (1983). Bayes Theory. Springer, New York.CrossRefGoogle Scholar
Hesselager, O. (1993). A class of conjugate priors with applications to excess-of-loss reinsurance. ASTIN Bulletin 23, 77–90.CrossRefGoogle Scholar
Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.CrossRefGoogle Scholar
Klugman, S.A. (1992). Bayesian Statistics in Actuarial Science. Kluwer, Boston.CrossRefGoogle Scholar
McNeil, A.J. (1997). Estimating the tails of loss severity distributions using extreme value theory. ASTIN Bulletin 27, 117–137.CrossRefGoogle Scholar
Pickands, J. III (1994). Bayes quantile estimation and threshold selection for the generalized Pareto family. In: Extreme Value Theory and Applications, 123–138, Galambos, J.et al. (eds.), Kluwer, Dordrecht.CrossRefGoogle Scholar
Reiss, R.-D. (1989). Statistical inference based on large claims via Poisson approximation. Part II: Poisson process approach. Blätter DGVM XIX, 123–128.Google Scholar
Reiss, R.-D. (1993). A Course on Point Processes. Springer, New York.CrossRefGoogle Scholar
Reiss, R.-D. and Thomas, M. (1997). Statistical Analysis of Extreme Values. Birkhäuser, Basel (with Xtremes on CD-ROM).CrossRefGoogle Scholar
Rytgaard, M. (1990). Estimation in the Pareto distribution. ASTIN Bulletin 20, 201–216.CrossRefGoogle Scholar
Schnieper, R. (1993). Praktische Erfahrungen mit Grossschadenverteilungen. Mitteil. Schweiz. Verein Versicherungsmath., 149–165.Google Scholar