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LAN OF EXTREME ORDER STATISTICS

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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*_{beta} : *beta* \in *Theta*}. We establish
local asymptotic normality (LAN) of the loglikelihood process
pertaining to the vector (*Z*_{n-i+1 : n})^{k}_{i=1} of the upper
*k* = *k*(*n*) \to _{n \to \infty} \infty order statistics in the sample, if the family
{*F*_{beta} : *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 *k*th-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 *beta* 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.

**Source**
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