Generalizations of the Hill estimator - asymptotic versus finite sample behaviour

The main goal of this paper is to present generalized Hill estimators parametrized in a positive real alpha (and equal to the Hill estimator when alpha = 1), which are asymptotically more efficient than the Hill estimator for a large region of values of alpha for any point of the (gamma,rho)-plane, where gamma > 0 is the tail index, related to the heaviness of the tail 1 - F of the underlying model F, and rho less than or equal to 0 is the second-order parameter, related to the rate of convergence of maximum values, linearly normalized, towards its limit. The practical validation of asymptotic results for small finite samples is done by means of simulation techniques in Frechet and Burr models, and some indications are provided on the choice of alpha. (C) 2001 Elsevier Science B.V. All rights reserved. MSG: 62G20; 62G30.