Abstract.    Consider an iid sample Z₁, ···, 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 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 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.

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