Optimal rates of convergence for estimates of the extreme value index

Hall and Welsh established the best attainable rate of convergence for estimates of a positive extreme value index gamma under a certain second order condition implying that the distribution function of the maximum of n random variables converges at an algebraic rate to the pertaining extreme value distribution. As a first generalization, we obtain a surprisingly sharp bound on the estimation error if gamma is still assumed to be positive, but the rate of convergence of the maximum may be nonalgebraic. This result allows a more accurate evaluation of the asymptotic performance of an estimator for gamma than the Hall and Welsh theorem. For example, it is proved that the Hill and the Pickands estimators achieve the optimal rate, but only the Hill estimator attains the sharp bound. Finally, an analogous result is derived for a general, not necessarily positive, extreme value index. In this situation it turns out that location invariant estimators show the best performance.