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Heavy-Tailed Distributions and Rating

Published online by Cambridge University Press:  29 August 2014

J. Beirlant ,
G. Matthys  and
G. Dierckx
J. Beirlant
Affiliation:
University Center of Statistics, Katholieke Universiteit Leuven
G. Matthys
Affiliation:
University Center of Statistics, Katholieke Universiteit Leuven
G. Dierckx
Affiliation:
University Center of Statistics, Katholieke Universiteit Leuven
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Abstract

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In this paper we consider the problem raised in the Astin Bulletin (1999) by Prof. Benktander at the occasion of his 80th birthday concerning the choice of an appropriate claim size distribution in connection with reinsurance rating problems. Appropriate models for large claim distributions play a central role in this matter. We review the literature on extreme value methodology and consider its use in reinsurance. Whereas the models in extreme-value methods are non-parametric or semi-parametric of nature, practitioners often need a fully parametric model for assessing a portfolio risk both in the tails and in more central portions of the claim distribution. To this end we propose a parametric model, termed the generalised Burr-gamma distribution, which possesses such flexibility. Throughout we consider a Norwegian fire insurance portfolio data set in order to illustrate the concepts. A small sample simulation study is performed to validate the different methods for estimating excess-of-loss reinsurance premiums.

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 31 , Issue 1 , May 2001 , pp. 37 - 58
Copyright
Copyright © International Actuarial Association 2001

References

1.Beirlant, J., Teugels, J.L. and Vynckier, P. (1996) Practical Analysis of Extreme Values. Leuven University Press.Google Scholar
2.Beirlant, J., Teugels, J.L. and Vynckier, P. (1996) Tail index estimation, Pareto quantile plots and regression diagnostics. J. Am. Statist. Assoc., 91, 1659–1667.Google Scholar
3.Beirlant, J., Dierckx, G., Goegebeur, Y. and Matthys, G. (1999) Tail index estimation and an exponential regression model. Extremes, 2, 177–200.CrossRefGoogle Scholar
4.Beirlant, J., Teugels, J.L. and Vynckier, P. (1996) Excess functions and the estimation of the extreme value index. Bernoulli, 2, 293–318.CrossRefGoogle Scholar
5.Beirlant, J., De Waal, D.J. and Teugels, J.L. (2000) The generalized Burr-gamma family of distributions with applications in extreme value analysis. To appear in: Proceedings of the 4th Conference on Limit Theorems in Probability and Statistics of the J. Bolyai Soc.Google Scholar
6.Benktander, G. and Segerdahl, C.O. (1960) On the analytical representation of claim distributions with special reference to excess-of-loss reinsurance. Trans. 16-th Intern. Congress Actuaries, 626–646.Google Scholar
7.Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
8.Coles, S.G. and Powell, E.A. (1996) Bayesian methods in extreme value modelling: a review and new developments. Int. Statist. Rev., 64, 119–136.CrossRefGoogle Scholar
9.Danielsson, J., de Haan, L., Peng, L. and De Vries, C.G. (1997) Using a bootstrap method to choose the sample fraction in tail index estimation. Technical report, Erasmus University, Rotterdam.Google Scholar
10.Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989) A moment estimator for the index of an extreme value distribution. Ann. Statist., 17, 1833–1855.Google Scholar
11.Dierckx, G. (2000) Estimation of the extreme value index. Ph.D. Thesis, Dept. of Mathematics, K.U. Leuven.Google Scholar
12.Drees, H. and Kaufmann, E. (1998) Selecting the optimal sample fraction in univariate extreme value estimation. Stoch. Proc. Appl., 75, 149–172.CrossRefGoogle Scholar
13.Embrechts, P., Klüppelberg, C. and Mikosch, Th. (1997) Modelling Extremal Events for Insurance and Finance. Springer Verlag, Berlin Heidelberg New York.CrossRefGoogle Scholar
14.Feuerverger, A. and Hall, P. (1999) Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Statist., 27, 760–781.CrossRefGoogle Scholar
15.Hill, B.M. (1975) A simple approach to inference about the tail of a dsitribution. Ann. Statist., 3, 1163–1174.CrossRefGoogle Scholar
16.Hosking, J.R.M. and Wallis, J.R. (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29, 339–349.CrossRefGoogle Scholar
17.Kratz, M. and Resnick, S. (1996) The qq-estimator of the index of regular variation. Commun. Statist., Stoch. Models, 12, 699–724.CrossRefGoogle Scholar
18.Mason, D.M. (1982) Laws of large numbers for sums of extreme values. Ann. Prob., 10, 756–764.CrossRefGoogle Scholar
19.McNeil, A. (1997) Estimating the tails of severity distributions using extreme value theory. Astin Bulletin.Google Scholar
20.Pickands, J. III (1975) Statistical inference using extreme order statistics. Ann. Statist., 3, 119–131.Google Scholar
21.Resnick, S.I. and Starica, C. (1997) Smoothing the Hill estimator. Adv. Appl. Probab., 29, 271–293.CrossRefGoogle Scholar
22.Rootzén, H. and Tajvidi, N. (1997) Extreme value statistics and wind storm losses: a case study. Scand. Act. J., 70–94.CrossRefGoogle Scholar
23.Smith, R.L. (1987) Estimating tails of probability distributions. Ann. Statist., 15, 1174–1207.CrossRefGoogle Scholar