Improving Second Order Reduced Bias Extreme Value Index Estimation

Abstract

Classical extreme value index estimators are known to be quite sensitive to the number k of top order statistics used in the estimation. The recently developed second order reduced-bias estimators show much less sensitivity to changes in k. Here, we are interested in the improvement of the performance of reduced-bias extreme value index estimators based on an exponential second order regression model applied to the scaled log-spacings of the top k order statistics. In order to achieve that improvement, the estimation of a “scale” and a “shape” second order parameters in the bias is performed at a level k1 of a larger order than that of the level k at which we compute the extreme value index estimators. This enables us to keep the asymptotic variance of the new estimators of a positive extreme value index γ equal to the asymptotic variance of the Hill estimator, the maximum likelihood estimator of γ, under a strict Pareto model. These new estimators are then alternatives to the classical estimators, not only around optimal and/or large levels k, but for other levels too. To enhance the interesting performance of this type of estimators, we also consider the estimation of the “scale” second order parameter only, at the same level k used for the extreme value index estimation. The asymptotic distributional properties of the proposed class of γ-estimators are derived and the estimators are compared with other similar alternative estimators of γ recently introduced in the literature, not only asymptotically, but also for finite samples through Monte Carlo techniques. Case-studies in the fields of finance and insurance will illustrate the performance of the new second order reduced-bias extreme value index estimators.