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Estimating catastrophic quantile levels for heavy-tailed distributions. (English) Zbl 1188.91237

Summary: Estimation of the occurrence of extreme events is of prime interest for actuaries. Heavy-tailed distributions are used to model large claims and losses. Within this setting we present a new extreme quantile estimator based on an exponential regression model that was introduced by A. Feuerverger and P. Hall [Ann. Stat. 27, No. 2, 760–781 (1999: Zbl 0942.62059)] and J. Beirlant et al. [Extremes 2, 177 (1999)]. We also discuss how this approach is to be adjusted in the presence of right censoring. This adaptation can also be linked to robust quantile estimation as this solution is based on a Winsorized mean of extreme order statistics which replaces the classical Hill estimator. We also propose adaptive threshold selection procedures for I. Weissman’s [J. Am. Stat. Assoc. 73, 812–815 (1978; Zbl 0397.62034)] quantile estimator which can be used both with and without censoring. Finally some asymptotic results are presented, while small sample properties are compared in a simulation study.

MSC:

91G70 Statistical methods; risk measures
62P05 Applications of statistics to actuarial sciences and financial mathematics
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
62E20 Asymptotic distribution theory in statistics
91B30 Risk theory, insurance (MSC2010)
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[1] Beirlant, J.; Dierckx, G.; Goegebeur, Y.; Matthys, G., Tail index estimation and exponential regression model, Extremes, 2, 177-200, (1999) · Zbl 0947.62034
[2] Beirlant, J.; Dierckx, G.; Guillou, A.; Stărică, C., On exponential representations of log-spacings of extreme order statistics, Extremes, 5, 157-181, (2002) · Zbl 1036.62040
[3] Beirlant, J.; Guillou, A., Pareto index estimation under moderate right censoring, Scandinavian actuarial journal, 2, 111-125, (2001) · Zbl 0979.91047
[4] Beirlant, J., Matthys, G., 2000. Adaptive threshold selection in tail index estimation. In: Embrechts, P. (Ed.), Extremes and Integrated Risk Management. Risk Books, London, pp. 37-49.
[5] Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press. · Zbl 0617.26001
[6] de Haan, L.; Rootzén, H., On the estimation of high quantiles, Journal of statistical planning and inference, 35, 1-13, (1993) · Zbl 0770.62026
[7] Deheuvels, P.; Haeusler, E.; Mason, D.M., Almost sure convergence of the Hill estimator, Mathematical Proceedings of the Cambridge philosophical society, 104, 371-381, (1988) · Zbl 0664.62023
[8] Delafosse, E.; Guillou, A., Almost sure convergence of a tail index estimator in the presence of censoring, Comptes rendus de l’académie des sciences de Paris, 335, 1-6, (2002) · Zbl 1140.60314
[9] Drees, H., On smooth statistical tail functionals, Scandinavian journal of statistics, 25, 187-210, (1998) · Zbl 0923.62032
[10] Drees, H., Extreme quantile estimation for dependent data, with applications to finance, Bernoulli, 9, 617-657, (2003) · Zbl 1040.62077
[11] Drees, H.; Kaufmann, E., Selecting the optimal sample fraction in univariate extreme value estimation, Stochastic processes and their applications, 75, 149-172, (1998) · Zbl 0926.62013
[12] Feuerverger, A.; Hall, P., Estimating a tail exponent by modelling departure from a Pareto distribution, Annals of statistics, 27, 760-781, (1999) · Zbl 0942.62059
[13] Grazier, K.L., G’Sell Associates, 1997. Group Medical Insurance Large Claims Database and Collection. SOA Monograph M-HB97-1. Society of Actuaries, Schaumburg, IL.
[14] Haeusler, E.; Teugels, J.L., On asymptotic normality of hill’s estimator for the exponent of regular variation, Annals of statistics, 13, 743-756, (1985) · Zbl 0606.62019
[15] Hill, B.M., A simple general approach to inference about the tail of a distribution, Annals of statistics, 3, 1163-1174, (1975) · Zbl 0323.62033
[16] Smith, R.L., Estimating tails of probability distributions, Annals of statistics, 15, 1174-1207, (1987) · Zbl 0642.62022
[17] Thépaut, A., Une nouvelle forme de réassurance. le traité d’excédent du coût moyen relatif ecomor, Bulletin trimestriel de l’institut des actuaires français, 49, 273, (1950) · Zbl 0038.30103
[18] Weissman, I., Estimation of parameters and large quantiles based on the k largest observations, Journal of American statistical association, 73, 812-815, (1978) · Zbl 0397.62034
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