## The Annals of Probability

### Inequalities for Distributions with Given Marginals

Andre H. Tchen

#### Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176994668

Digital Object Identifier
doi:10.1214/aop/1176994668

Mathematical Reviews number (MathSciNet)
MR577318

Zentralblatt MATH identifier
0459.62010

JSTOR