Revisiting the role of the jackknife methodology in the estimation of a positive tail index

Abstract

In this article, and in a context of regularly varying tails, we analyze a generalization of the classical Hill estimator of a positive tail index. The members of this general class of estimators are not asymptotically more eficient than the original one, the Hill estimator. We thus propose a class of generalized Jackknife estimators associated with any two members of the first class. They enable the reduction of the main component of bias of the Hill estimator and are dependent of a tuning parameter, which is adequately chosen through an asymptotic variance minimization criterion. These new estimators are compared with the Hill estimator, both asymptotically and for finite samples and, when the underlying distribution is in Hall's class of models, they really improve on the well-known, bias-variance, trade-off characteristic of the Hill estimator.