A problem on stability of order statistics

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Abstract

Let X1, . . . , Xn, be a sequence of independent and identically distributed random variables with a continuous distribution function F and let Xn-kn+1 be the corresponding knth largest order statistics. Gather and Tomkins (J. Statist. Plann. Inference (1995), 175-183) proved that the condition (C) lim supx→ωr (1 - F (x + ε))/(1 - F (x)) < 1, where ωr := sup{x: F (x) < 1}, is sufficient for every sequence of upper-intermediate order statistics being absolutely stable. In this paper we prove that the condition (C) is also necessary, which solves an open problem by Gather and Tomkins (J. Statist. Plann. Inference (1995), 175-183). As an application of this result we give a criterion for every sequence of order statistics being absolutely stable.

Original languageEnglish (US)
Pages (from-to)21-25
Number of pages5
JournalJournal of Statistical Planning and Inference
Volume64
Issue number1
StatePublished - Oct 30 1997
Externally publishedYes

Keywords

  • Order statistics
  • Stability

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