On the Estimation of the Second Order Parameter for Heavy-Tailed Distributions

Abstract

The extreme-value index γ is an important parameter in extreme-value theory since it controls the first order behavior of the distribution tail. In the literature, numerous estimators of this parameter have been proposed especially in the case of heavy-tailed distributions, which is the situation considered here. Most of these estimators depend on the k largest observations of the underlying sample. Their bias is controlled by the second order parameter ρ. In order to reduce the bias of γ’s estimators or to select the best number k of observations to use, the knowledge of ρ is essential. In this paper, we propose a simple approach to estimate the second order parameter ρ leading to both existing and new estimators. We establish a general result that can be used to easily prove the asymptotic normality of a large number of estimators proposed in the literature or to compare different estimators within a given family. Some illustrations on simulations are also provided.