Electronic Resource
Springer
Annals of the Institute of Statistical Mathematics
47 (1995), S. 693-717
ISSN:
1572-9052
Keywords:
Extreme order statistics
;
local asymptotic normality
;
central sequence
;
generalized Pareto distributions
;
asymptotic sufficiency
;
optimal tests
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 ) i=1 k 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01856542
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