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.

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

“…As mentioned before, as |ρ| < 1 in all simulated models, we have chosen the tuning parameter τ = 0. When γ + ρ = 0, the PORT-ML estimator is a second order reduced bias estimator and we therefore need to consider heuristic adaptive choices of the threshold, as in lines of Gomes et al (2008b). The adaptive choices considered in this work are:k ML 01 :=k MP 0 andk ML 02 := n − 1, with indicators,…”

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].…”