Asymptotic comparison of the mixed moment and classical extreme value index estimators

Abstract

A new promising extreme value index estimator, the mixed-moment (MM) estimator, has been recently introduced in the literature. This estimator uses not only the first moment of the top excesses of the log-observations in the sample, the basis of the classical Hill and moment estimators, but also the first moment of another type of statistics, dependent on quotients of top order statistics. In this paper we shall compare, asymptotically at optimal levels, the MM estimator with the classical Hill, the moment and the usually denoted “maximum likelihood” extreme value index estimator, associated to an approximation for the excesses over a high observation provided by the generalized Pareto distribution. Again, the MM estimator cannot always dominate the alternatives, but its asymptotic performance is quite interesting.