Asymptotic Properties of the Block-Type Statistics

Abstract

Extreme value theory is an issue extensively applied in many different fields. One of the central points of this theory is the estimation of a positive extreme value index. In this paper we introduce a new family of block type statistics related to this estimation. A weak consistency of the introduced statistics is proved. A bivariate central limit theorem for newly introduced statistics is derived. We provide the new family of semi-parametric estimators for the positive extreme value index. Asymptotic normality of the introduced estimators is proved. It is shown that new estimators have better asymptotic performance comparing with several block-type estimators over the whole range of parameters presented in the second order regular variation condition. An application to the estimation of the positive valued extreme value index for several real data sets is provided.