LAN of extreme order statistics

Consider an iid sampleZ1,...,Zn with common distribution functionF on the real line, whose upper tail belongs to a parametric family {Fβ: β∈⊝}. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Zn−i+1∶n)i=1k of the upperk=k(n)→n→∞∞ order statistics in the sample, if the family {Fβ:β∈⊝} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZn−k+1∶n is the central sequence generating LAN. This implies thatZn−k+1∶n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter β can be based on the single order statisticZn−k+1∶n. The rate at whichZn−k+1∶n becomes asymptotically sufficient is however quite poor.

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