The Annals of Probability

Inequalities for Distributions with Given Marginals

Andre H. Tchen

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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.

Article information

Ann. Probab. Volume 8, Number 4 (1980), 814-827.

First available in Project Euclid: 19 April 2007

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Primary: 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory 62H20: Measures of association (correlation, canonical correlation, etc.)

Distributions with given marginals correlation measures of dependence concordance stochastic ordering inequalities bounds rearrangements


Tchen, Andre H. Inequalities for Distributions with Given Marginals. Ann. Probab. 8 (1980), no. 4, 814--827. doi:10.1214/aop/1176994668.

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