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Selecting the optimal sample fraction in univariate extreme value estimation. (English) Zbl 0926.62013

The paper deals with optimal estimation of the parameter of an asymptotic extreme value distribution (extreme value index \(\beta \)). A set of consistent estimators of the extreme value index has already been proposed in several previous papers. As a rule, the estimate of \(\beta \) is computed from the largest \(k_n\) order statistics. It has also been shown that the behaviour of the estimate depends on the selection of \(k_n\). The authors present an adaptive way of selection of the fraction \(k_n\) and they prove that their choice leads to an optimal estimator of \(\beta \), in the sense of minimal asymptotic mean squared error. The results are proved for the case of the Hill estimator of \(\beta \) [B.M. Hill, Ann. Stat. 3, 1163-1174 (1975; Zbl 0323.62033); P. Hall and A.H. Welsh, ibid. 13, 331-341 (1985; Zbl 0605.62033)], but the approach can also be modified for the adaptive selection of the optimal fraction of the sample for a rather broad class of estimators of the extreme value index.
The approach is based on the results proved in Lemma 1 and its Corollary 1 showing that the maximum of random fluctuations of extreme value index estimates is of the order of the iterated logarithm. The advantage of the method consists in the selection of the optimal fraction of the sample results from comparisons of the estimates of \(\beta\) for different fractions, while some more traditional approaches use additional properties of the underlying distribution function.
The paper contains also a simulation study showing the behaviour of the estimator in the case of moderate sample sizes and for several types of underlying probability distributions.
Reviewer: P.Volf (Praha)

MSC:

62F10 Point estimation
62F35 Robustness and adaptive procedures (parametric inference)
62F12 Asymptotic properties of parametric estimators
62G30 Order statistics; empirical distribution functions
62E20 Asymptotic distribution theory in statistics
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