Asymptotically best linear unbiased tail estimators under a second-order regular variation condition

For regularly varying tails, estimation of the index of regular variation, or tail index, gamma, is often performed through the classical Hill estimator, a statistic strongly dependent on the number k of top order statistics used, and with a high asymptotic bias as k increases, unless the underlying model is a strict Pareto model. First, on the basis of the asymptotic structure of Hill's estimator for different k-values, we propose "asymptotically best linear (BL) unbiased" estimators of the tail index. A similar derivation on the basis of the log-excesses and of the scaled log-spacings is performed and the adequate weights for the largest log-observations are provided. The asymptotic behaviour of those estimators is derived, and they are compared with other alternative estimators, both asymptotically and for finite samples. As an overall conclusion we may say that even asymptotic equivalent estimators may exhibit very diversified finite sample properties. (c) 2004 Elsevier B.V. All rights reserved.