Characterization of the exponential distribution by properties of the difference \(X_{k+s:n}-X_{k:n}\) of order statistics
type: Item
instance of: scholarly article
title: Characterization of the exponential distribution by properties of the difference \(X_{k+s:n}-X_{k:n}\) of order statistics
zbMATH Open document ID: 0883.60012
DOI: 10.4171/ZAA/758
author: O. Diekmann
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1998-03-04T00:00:00Z
review text: Summary: Let \(X_1,X_2,\dots,X_n\) be independent and identically distributed random variables subject to a continuous distribution function \(F\), let \(X_{1:n},X_{2:n},\dots,X_{n:n}\) be the corresponding order statistics, and write \[ P(X_{k+s:n}- X_{k:n}\geq x)= P(X_{s:n-k}\geq x)\qquad (x\geq 0),\tag{0} \] where \(n\), \(k\) and \(s\) are fixed integers with \(k+s\leq n\). It is an old question if condition (0) implies that \(F\) is of exponential type. The first and second authors [Metrika 41, No. 1, 1-19 (1994; Zbl 0793.62011)] showed among others that condition (0) can be greatly relaxed; namely, it can be replaced by asymptotic relations (either as \(x\to\infty\) or \(x\downarrow 0\)) to derive this very result. Using a theorem on integrated Cauchy functional equations and in essential way a result of the above quoted paper, we find now a more elegant and deeper theorem on this subject. The case of lattice distributions is also considered and some new problems are stated.
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 45A05
zbMATH DE Number: 1011109
zbMATH Keywords: exponential distributions
zbMATH Keywords: integrated Cauchy functional equations
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/758
OpenAlex ID: W2076149935
description: scientific article; zbMATH DE number 1011109