Efficiency of convex combinations of pickands estimator of the extreme value index

Abstract

Consider a distribution function F in the domain of attraction of an extreme value distribution G β with unknown extreme value index . An appealing nonparametric estimate of β is the Pickands estimator , which is based on the 4m largest observations in a sample of size n generated independently according to F. If F satisfies a von Mises condition with rapidly decreasing remainder term, we can establish asymptotic normality of convex combinations of Pickands estimate. With the asymptotically optimal p = p opt∊[0,1] minimizing the limiting variance of Pickands estimator is then clearly outperformed by the convex combination . As popt depends on β, a data-driven version is plugged into , with the resulting estimate being asymptotically as good as . Simulations demonstrate the superiority of the convex combination over the Pickands estimate already for moderate sample sizes n.