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

Dekkers, A. L. M.; Einmahl, J. H. J.; De Haan, L.

Ann. Stat. 17, No. 4, 1833-1855 (1989).

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

Keywords:

quantile estimation; extreme-value distribution; Hill’s estimator; index of a distribution function; regularly varying tail; Consistency; asymptotic normality; endpoint estimation

Citations:

Zbl 0323.62033