LAN OF EXTREME ORDER STATISTICS

MICHAEL FALK

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

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

Abstract. Consider an iid sample Z_{1}, …, Z_{n} with common distribution function F on the real line, whose upper tail belongs to a parametric family {F_{β} : β ∈ Θ}. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector (Z_{n−i+1:n})_{i=1}^{k} of the upper k = k(n) →_{n→∞} ∞ order statistics in the sample, if the family {F_{β} : β ∈ Θ} 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 Z_{n−k+1:n} is the central sequence generating LAN. This implies that Z_{n−k+1:n} is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter β can be based on the single order statistic Z_{n−k+1:n}. The rate at which Z_{n−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.