Tail index estimation for heavy tails: accommodation of bias in the excesses over a high threshold

Abstract

In statistics of extremes, inference is often based on the excesses over a high random threshold. Those excesses are approximately distributed as the set of order statistics associated to a sample from a generalized Pareto model. We then get the so-called "maximum likelihood" estimators of the tail index γ . In this paper, we are interested in the derivation of the asymptotic distributional properties of a similar "maximum likelihood" estimator of a positive tail index γ , based also on the excesses over a high random threshold, but with a trial of accommodation of bias in the Pareto model underlying those excesses. We next proceed to an asymptotic comparison of the two estimators at their optimal levels. An illustration of the finite sample behaviour of the estimators is provided through a small-scale Monte Carlo simulation study.

The PORTSEA (Portuguese School of Extremes and Applications) and a few personal scientific achievements

“…From 2005 onwards, the adequate estimation of second order parameters, essentially due to developments achieved in articles referred to in 3.5.4, allowed us to maintain the variance and eliminate the dominant component of asymptotic bias, drastically improving the behaviour of the estimators for all k. Considering only the EVI-estimation, and essentially for heavy tails, details about these new MVRB estimation methods can be seen in: [65,206,233,255], with the accommodation of bias performed in the excesses over a high level; [189], with bias accommodation performed in the weighted excesses of the top log-observations; [224]; [62], under a third-order framework; [44,46,54,180]; [177] and [61], both for GMs and already referred in 3.4.7; [34,211,212], with comparisons of a large diversity of competitive estimators; [179]; [249], in which we draw attention to the need to debate the topic of 'efficiency vs robustness' in statistics of extremes; [38,68].…”

Location invariant Weiss-Hill estimator

“…For more related work on extreme value index estimation, see Beirlant et al (1996), de Haan andPeng (1998), Peng (1998a), Martins (2002, 2004), Peng and Qi (2006a, b), Gomes et al (2008a, b), Gomes and Henriques Rodrigues (2008), Li et al (2008Li et al ( , 2010 and Qi (2010).…”