Local asymptotic normality of intermediate and central order statistics |
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Authors: | Michael Falk |
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Affiliation: | Mathematisch-Geographische , Kathohsche Universit?t Eichst?tt , Fakult?t, Eichst?tt, 85071 |
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Abstract: | Consider n independent random variables Zi,…, Zn on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = Fθ(t),t ≥ x0, where θ ∈ ? ? R d. A necessary and sufficient condition for the family Fθ, θ ∈ ?, is established such that the k-th largest order statistic Zn?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Zn?i+1:n)1 i=kof the k largest order statistics. This is achieved for k = k(n)→n→∞∞ with k/n→n→∞ 0. In the case of vectors of central order statistics ( Zr:n, Zr+1:n,…, Zs:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Zm:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only |
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Keywords: | LAN central sequence asymptotically optimal test efficient estimator sequences generalized Pareto distribution |
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