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Stability of Order Statistics under Dependence

โœ Scribed by Tomasz Rychlik


Book ID
110330484
Publisher
Springer Japan
Year
2001
Tongue
English
Weight
968 KB
Volume
53
Category
Article
ISSN
0020-3157

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If X 1 , ..., X n are random variables we denote by X (1) X (2) ... X (n) their respective order statistics. In the case where the random variables are independent and identically distributed, one may demonstrate very strong notions of dependence between any two order statistics X (i) and X ( j) . I

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For r >/1, let Mnr be the rth largest of {X1,... ,X,}, where XI,X2 .... are i.i.d, random variables with common distribution function F such that F(x)< 1 for all x. Define #(x) =F 1(1 -x -~), x>~ 1. {M,r,n>~r} is called regularly stable if Mnr/12(n)--~ 1 in probability and (\*)#(n t)/#(n) ~ h(t)< ve