A class of asymptotically unbiased semi-parametric estimators of the tall index

In this paper we consider a class of consistent semi-parametric estimators of a positive tail index gamma, parameterized in a tuning or control parameter alpha. Such a control parameter enables us to have access, for any available sample, to an estimator of gamma with a null dominant component of asymptotic bias, and with a reasonably flat Mean Squared Error pattern, as a function of kappa, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relatively to the classical Hill estimator, provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator.