A heuristic adaptive choice of the threshold for bias-corrected Hill estimators

Abstract

We shall deal with specific classes of the second-order reduced bias extreme value index estimators, devised for heavy tails. In those classes, the second-order parameters in the bias are estimated at a level k 1 of a larger order than that of the level k at which we compute the extreme value index estimator, and by doing this, it is possible to keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator, the maximum-likelihood estimator of the extreme value index γ, under a strict Pareto model. On the basis of an adequate pair of this type of extreme value index estimators, we also provide a heuristic adaptive choice of the threshold in reduced bias estimation and we proceed to an intensive computer simulation that enables us to study, through Monte-Carlo techniques, the behavior of the non-adaptive and adaptive proposed estimators. An illustration of the behavior of these estimators for sets of real data in the fields of finance and insurance is also provided.