Abstract
Consider an iid sampleZ 1,...,Z n with common distribution functionF on the real line, whose upper tail belongs to a parametric family {F β: β∈⊝}. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Z n−i+1∶n ) of the upperk=k(n)→ n→∞∞ order statistics in the sample, if the family {F β: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 statisticZ n−k+1∶n is the central sequence generating LAN. This implies thatZ n−k+1∶n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter β can be based on the single order statisticZ n−k+1∶n . The rate at whichZ n−k+1∶n becomes asymptotically sufficient is however quite poor.
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