A problem on stability of order statistics

Yongcheng Qi

doi:10.1016/s0378-3758(96)00206-6

A problem on stability of order statistics

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Abstract

Let X₁, . . . , X_n, be a sequence of independent and identically distributed random variables with a continuous distribution function F and let X_n-kn+1 be the corresponding k_nth largest order statistics. Gather and Tomkins (J. Statist. Plann. Inference (1995), 175-183) proved that the condition (C) lim sup_x→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 language	English (US)
Pages (from-to)	21-25
Number of pages	5
Journal	Journal of Statistical Planning and Inference
Volume	64
Issue number	1
DOIs	https://doi.org/10.1016/s0378-3758(96)00206-6
State	Published - Oct 30 1997
Externally published	Yes

Bibliographical note

Funding Information:
1 Supported by the Netherlands Foundation for Mathematics SMC with a grant from the Netherlands Organization for Scientific Research NWO. The author would like to thank Prof. L. de Haan for his hospitality.

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

Keywords

Order statistics
Stability

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N2 - 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.

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A problem on stability of order statistics

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