Estimating catastrophic quantile levels for heavy-tailed distributions

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 Feuerverger and Hall [Ann. Stat. 27 (1999) 760] and Beirlant et al. [Extremes 2 (1999) 177]. 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 Weissman’s [J. Am. Stat. Assoc. 73 (1978) 812] 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.