A Semiparametric Bootstrap Technique for Simulating Extreme Order Statistics

Abstract

We propose a technique for simulating the joint distribution of the j largest order statistics of a very large sample. We assume that the parent population is in the domain of attraction of the Type 1 (Gumbel) extreme value distribution. The bootstrap variates are generated by resampling the normalized spacings of the k largest observed values in the original data where k is larger than j. We compare the bootstrap distribution to the fitted extremal distribution of Weissman. Both distributions have the same means, conditional on the k largest observed values in the data set. If k is large and the normalized spacings behave as independent and identically distributed exponential random variables then the bootstrap variates behave as though sampled from the extremal distribution. We propose several procedures for estimating k and give a numerical example.

Key Words:

• Exponential distribution
• Extreme value distribution
• Gini statistic
• Spacings