A simple generalisation of the Hill estimator

Abstract

The classical Hill estimator of a positive extreme value index (EVI) can be regarded as the logarithm of the geometric mean, or equivalently the logarithm of the mean of order p=0, of a set of adequate statistics. A simple generalisation of the Hill estimator is now proposed, considering a more general mean of order p≥0 of the same statistics. Apart from the derivation of the asymptotic behaviour of this new class of EVI-estimators, an asymptotic comparison, at optimal levels, of the members of such class and other known EVI-estimators is undertaken. An algorithm for an adaptive estimation of the tuning parameters under play is also provided. A large-scale simulation study and an application to simulated and real data are developed.