Nonparametric tail estimation using a double bootstrap method

Extreme value theory has led to the development of various statistical methods for nonparametric estimation of distribution tails. A common problem in all of these estimators is the choice of the number of extreme data that should be used in the estimation and the construction of confidence intervals on the estimator. In this paper, we outline a method that uses the nonparametric bootstrap for both problems. The bootstrap is twofold: (1) the first bootstrap is used to estimate the optimal number of extremes - in the mean square error sense - to be used for the tail index estimation as has been earlier suggested by Hall (1990, J. Multivariate Anal. 32 (1990) 177-203), and (2) the second bootstrap is used to obtain confidence intervals. The method has been applied to data generated by Monte Carlo simulation for a variety of distributions and on this basis the performance of the method will be assessed. (C) 1999 Elsevier Science B.V. All rights reserved.