Minimum-variance reduced-bias estimation of the extreme value index: A theoretical and empirical study

In extreme value (EV) analysis, the EV index (EVI), ξ, is the primary parameter of extreme events. In this work, we consider ξ positive, that is, we assume that F is heavy tailed. Classical tail parameters estimators, such as the Hill, the Moments, or the Weissman estimators, are usually asymptotically biased. Consequently, those estimators are quite sensitive to the number of upper order statistics used in the estimation. Minimum-variance reduced-bias (RB) estimators have enabled us to remove the dominant component of asymptotic bias without increasing the asymptotic variance of the new estimators. The purpose of this paper is to study a new minimum-variance RB estimator of the EVI. Under adequate conditions, we prove their nondegenerate asymptotic behavior. Moreover, an asymptotic and empirical comparison with other minimum-variance RB estimators from the literature is also provided. Our results show that the proposed new estimator has the potential to be very useful in practice.