## A moment estimator for the index of an extreme-value distribution.(English)Zbl 0701.62029

Let $$X_ 1,X_ 2,..$$. be a sequence of i.i.d. random variables. Suppose for some constants $$a_ n>0$$ and $$b_ n$$, $$n=1,2,...$$, and some $$- \infty <\gamma <\infty:$$ $\lim_{n\to \infty}P\{a_ n^{-1}(\max (X_ 1,...,X_ n)-b_ n)\leq x\}=G_{\gamma}(x),-\infty <x<\infty,$ where $$G_{\gamma}(x)=\exp (-(1+\gamma x)^{-1/\gamma})$$ is the extreme-value distribution.
The authors consider the estimation problem for the index $$\gamma$$. They extend Hill’s estimator for the index of a distribution function with regularly varying tail [B. M. Hill, ibid. 3, 1163-1174 (1975; Zbl 0323.62033)] to an estimate for $$\gamma$$. Consistency and asymptotic normality are proved. The authors also use the estimator for quantile and endpoint estimation.
Reviewer: W.Dziubdziela

### MSC:

 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions

Zbl 0323.62033
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