Optimal Rates of Convergence for Estimates of the Extreme Value Index

Hall and Welsh (1984) established the best attainable rate of convergence for estimates of a positive extreme value index 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 rst generalization we obtain a surprisingly sharp bound on the estimation error if is still assumed to be positive, but the rate of convergence of the maximum may be non-algebraic. This result allows a more accurate evaluation of the asymptotic performance of an estimator for than Hall's and Welsh's theorem. For example, it is proven that the Hill and the Pickands estimator 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.