Resampling methodologies and reliable tail estimation

M. Ivette Gomes; Fernanda Figueiredo; M. João Martins; Manuela Neves

M. Ivette Gomes Centro de Estatística e Aplicações (CEA), Faculdade de Ciências, Universidade de Lisboa
Fernanda Figueiredo CEA, Universidade de Lisboa, and Faculdade de Economia, Universidade do Porto
M. João Martins Centro de Estudos Florestais and Instituto Superior de Agronomia, Universidade de Lisboa
Manuela Neves CEA, and Instituto Superior de Agronomia, Universidade de Lisboa

Keywords: Bootstrap and jackknife methodologies, Semi-parametric estimation, Statistics of extremes

Abstract

Resampling methodologies, like the generalised jackknife and the bootstrap are important
tools for a reliable semi-parametric estimation of parameters of extreme or even rare events. Among these parameters we mention the extreme value index, the primary parameter in statistics of extremes, and the extremal index, a measure of clustering of extreme events. Most of the semi-parametric estimators of these parameters show the same type of behaviour: nice asymptotic properties, but a high variance for small k, the number of upper order statistics used in the estimation, a high bias for large k, and the need for an adequate choice of k. After a brief reference to some estimators of the aforementioned parameters and their asymptotic properties we present algorithms for an adaptive reliable estimation of the extreme value and extremal index.

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