A Couple of Non Reduced Bias Generalized Means in Extreme Value Theory

Abstract

Lehmer’s mean-of-order p (Lp) generalizes the arithmetic mean, and Lp extreme value index (EVI)-estimators can be easily built, as a generalization of the classical Hill EVI-estimators. Apart from a reference to the asymptotic behaviour of this class of estimators, an asymptotic comparison, at optimal levels, of the members of such a class reveals that for the optimal (p, k) in the sense of minimal mean square error, with k the number of top order statistics involved in the estimation, they are able to overall outperform a recent and promising generalization of the Hill EVI-estimator, related to the power mean, also known as H¨older’s mean-of-order-p. A further comparison with other ‘classical’ non-reduced-bias estimators still reveals the competitiveness of this class of EVI-estimators.

Keywords

• heavy tails
• optimal tuning parameters
• semi-parametric estimation
• statistical extreme value theory