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BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION: A REVIEW
M. MANUELA NEVES
Dedicated to Nazar´e Lopes as a token of friendship
Abstract. This paper presents an overview of some applications of re- sampling computer intensive methodologies, like the bootstrap and the jackknife, in a reliable estimation of the extremal index. The extremal index, a measure of clustering of extreme events, is a key parameter in extreme value theory in a dependent set-up. Its estimation has been con- sidered by several authors but some di culties still remain. Most of the semi-parametric estimators of this parameter show the same type of be- haviour: 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 review of some estimators and their asymptotic properties, two computational procedures, the Block-Bootstrap and Jackknife-After-Bootstrap are ap- plied to improve the extremal index estimation. An adaptive choice of the block size for the block-bootstrap resampling is presented. A few results of a simulation study will illustrate the application of that choice.
1. Introduction and the scope of the paper
In Extreme Value Theory (EVT) we deal essentially with the estimation of parameters of extreme or rare events. A large number of applications in areas such as biology, environment, finance, hydrology and telecommunications, reveals the importance of adequate estimation procedures.
The classical assumption in EVT is that we have a set of independent and identically distributed (i.i.d.) random variables (r.v.’s), X1, . . . , Xn, from an
Accepted: 07 April 2015.
2010 Mathematics Subject Classification. Primary 62G09, 62E20, 60G10; Secondary
60G70.
Key words and phrases. Bootstrap and Jackknife, extremal index, semi-parametric esti-
mation, statistics of extremes.
The work was supported by National Funds through FCT – Fundac¸˜ao para a Ciˆencia e a
Tecnologia, project PEst-OE/MAT/UI0006/2014 (CEAUL).
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