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August, 1980 Inequalities for Distributions with Given Marginals
Andre H. Tchen
Ann. Probab. 8(4): 814-827 (August, 1980). DOI: 10.1214/aop/1176994668


An ordering on discrete bivariate distributions formalizing the notion of concordance is defined and shown to be equivalent to stochastic ordering of distribution functions with identical marginals. Furthermore, for this ordering, $\int\varphi dH$ is shown to be $H$-monotone for all superadditive functions $\varphi$, generalizing earlier results of Hoeffding, Frechet, Lehmann and others. The usual correlation coefficient, Kendall's $\tau$ and Spearman's $\rho$ are shown to be monotone functions of $H$. That $\int\varphi dH$ is $H$-monotone holds for distributions on $\mathbb{R}^n$ with fixed $(n - 1)$-dimensional marginals for any $\varphi$ with nonnegative finite differences of order $n$. Some related results are obtained. Stochastic ordering is preserved under certain transformations, e.g., convolutions. A distribution on $\mathbb{R}^\infty$ is constructed, making $\max(X_1,\cdots, X_n)$ stochastically largest for all $n$ when $X_i$ have given one-dimensional distributions, generalizing a result of Robbins. Finally an ordering for doubly stochastic matrices is proposed.


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Andre H. Tchen. "Inequalities for Distributions with Given Marginals." Ann. Probab. 8 (4) 814 - 827, August, 1980.


Published: August, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0459.62010
MathSciNet: MR577318
Digital Object Identifier: 10.1214/aop/1176994668

Primary: 62E10
Secondary: 62H05 , 62H20

Keywords: bounds , concordance , Correlation , distributions with given marginals , Inequalities‎ , measures of dependence , rearrangements , stochastic ordering

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 4 • August, 1980
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