A Pickands type estimator of the extreme value index (original) (raw)
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Due to the fact that for heavy tails the classical Hill estimator of a positive extreme value index is asymptotically biased, new and interesting alternative estimators have appeared in the literature. In this work we compare several classical estimators of the extreme value index based on moments of the upper order statistics. Since several alternative estimators have eventually a null asymptotic bias, for some heavy tailed models, the comparison is performed not only with the Hill and recent generalized means estimators but also with an asymptotically unbiased Hill estimator. The comparison study is performed asymptotically, under a third-order framework, and for finite samples, through a Monte Carlo simulation study.
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A large part of the theory of extreme value index estimation is developed for positive extreme value indices. The best known estimator for that case is the Hill estimator. This estimator can be considered to be either a moment estimator or a (quasi) maximum likelihood estimator and was generalized to a kernel-type estimator, still only valid for positive extreme value indices. The Hill estimator has been extended to a momenttype estimator valid for all extreme value indices. Also the quasi maximum likelihood estimators based on the generalized Pareto distribution, have been given for a restricted region of negative extreme value indices We derive kernel-type estimators valid for all real extreme value indices and compare their performance with the (generalized) moment estimator and (quasi) maximum likelihood estimator.
Computational and Mathematical Methods, 2020
In extreme value (EV) analysis, the EV index (EVI), , is the primary parameter of extreme events. In this work, we consider positive, that is, we assume that F is heavy tailed. Classical tail parameters estimators, such as the Hill, the Moments, or the Weissman estimators, are usually asymptotically biased. Consequently, those estimators are quite sensitive to the number of upper order statistics used in the estimation. Minimum-variance reduced-bias (RB) estimators have enabled us to remove the dominant component of asymptotic bias without increasing the asymptotic variance of the new estimators. The purpose of this paper is to study a new minimum-variance RB estimator of the EVI. Under adequate conditions, we prove their nondegenerate asymptotic behavior. Moreover, an asymptotic and empirical comparison with other minimum-variance RB estimators from the literature is also provided. Our results show that the proposed new estimator has the potential to be very useful in practice.