Bootstrap confidence intervals for tail indices

Abstract

One of the classical problems in extreme value statistics concerns the coice of the number of extremes used in estimation. It is known that there is a bias-variance trade-off when changing the threshold above which extremes are retained. A mean squared error approach is therefore often used as the governing criterion for picking an optimal level. Hill-type estimators for tail indices were introduced in Beirlant et al. (J. Amer. Statist. Assoc. (1996a); Bernoulli (1996b)) in an iterative fashion using the complete sample to estimate adaptively the optimal number of extremes to be used in the tail estimation problem. The methodology is based on the minimization of the asymptotic mean squared error. We propose a nonparametric bootstrap solution for the open problem of obtaining workable finite sample confidence intervals of these extreme value estimators. Monte Carlo simulation will be used to analyse the accurateness of coverage probabilities and to study the effect of bias.