Some partial orderings of exchangeable random variables by positive dependence
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.
scientific article; zbMATH DE number 3913465
scientific article