### MICHAEL FALK

Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt,
85071 Eichstätt, Germany

(Received March 10, 1994; revised December 16, 1994)

Abstract.    Consider an iid sample Z1, ···, Zn with common distribution function F on the real line, whose upper tail belongs to a parametric family {Fbeta : beta \in Theta}. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector (Zn-i+1 : n)ki=1 of the upper k = k(n) \to n \to \infty \infty order statistics in the sample, if the family {Fbeta : beta \in Theta} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, the kth-largest order statistic Zn-k+1 : n is the central sequence generating LAN. This implies that Zn-k+1 : n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter beta can be based on the single order statistic Zn-k+1 : n. The rate at which Zn-k+1 : n becomes asymptotically sufficient is however quite poor.

Key words and phrases:    Extreme order statistics, local asymptotic normality, central sequence, generalized Pareto distributions, asymptotic sufficiency, optimal tests.

Source ( TeX , DVI , PS )