Some partial orderings of exchangeable random variables by positive dependence
type: Item
instance of: scholarly article
title: Some partial orderings of exchangeable random variables by positive dependence
zbMATH Open document ID: 0572.62041
DOI: 10.1016/0047-259X(85)90088-0
author: Y. L. Tong
author: Moshe Shaked
published in: Journal of Multivariate Analysis
publication date: 1985-01-01T00:00:00Z
review text: The authors consider exchangeable random vectors \b{X}\(=(X_ 1,...,X_ n)\) whose entries are positively correlated. For such vectors they introduce a number of partial orderings in which \b{X} is ''bigger'' than \b{Y}\(=(Y_ 1,...,Y_ n)\) precisely when, in some sense or other, the \(X_ i\) are less positively correlated or ''more dispersed'' than the \(Y_ i\). For example, they declare that \b{X}\(\succ_ A\underline Y\) if \[ | \sum^{n}_{i=1}c_ iX_{(i)}| \geq^{st}| \sum^{n}_{i=1}c_ iY_{(i)}| \quad whenever\quad \sum^{n}_{i=1}c_ i=0, \] where \(X_{(i)}\) (resp. \(Y_{(i)})\) denotes the ith order statistic of \b{X} (resp. of \b{Y}), and where \(\geq^{st}\) denotes stochastic ordering, that is, \(U\geq^{st}V\) if \(P(U>a)\geq P(V>a)\) for every real number a. These partial orderings are interrelated and their basic properties obtained. For example, if \b{X}\(\succ_ A\underline Y\), then \((X_{i_ 1},...,X_{i_ k})\succ_ A(Y_{i_ 1},...,Y_{i_ k})\) whenever \(1\leq i_ 1<...<i_ k\leq n\). They are then shown to arise naturally in two fairly general probability models of multivariate analysis, models in which nearly all of the familiar multivariate distributions may appear. Applications to genetic selection and choice of sampling procedures are presented.
Mathematics Subject Classification ID: 62H05
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G09
zbMATH DE Number: 3913465
zbMATH Keywords: Cauchy
zbMATH Keywords: exponential
zbMATH Keywords: binomial
zbMATH Keywords: Poisson
zbMATH Keywords: gamma
zbMATH Keywords: Farlie-Gumbel-Morgenstern
zbMATH Keywords: positive dependence by mixture
zbMATH Keywords: majorization
zbMATH Keywords: partial orderings
zbMATH Keywords: order statistic
zbMATH Keywords: stochastic ordering
zbMATH Keywords: genetic selection
zbMATH Keywords: choice of sampling procedures
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3913465