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

Matthys, Gunther; Delafosse, Emmanuel; Guillou, Armelle; Beirlant, Jan

Insur. Math. Econ. 34, No. 3, 517-537 (2004).

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)

Keywords:

Pareto index; Extreme quantiles; Censoring; Bias reduction

Citations:

Zbl 0942.62059; Zbl 0397.62034

References:

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