The Annals of Statistics

Majorization in Multivariate Distributions

Albert W. Marshall and Ingram Olkin

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In case the joint density $f$ of $X = (X_1, \cdots, X_n)$ is Schur-concave (is an order-reversing function for the partial ordering of majorization), it is shown that $P(X \in A + \theta)$ is a Schur-concave function of $\theta$ whenever $A$ has a Schur-concave indicator function. More generally, the convolution of Schur-concave functions is Schur-concave. The condition that $f$ is Schur-concave implies that $X_1, \cdots, X_n$ are exchangeable. With exchangeability, the multivariate normal and certain multivariate "$t$", beta, chi-square, "$F$" and gamma distributions have Schur-concave densities. These facts lead to a number of useful inequalities. In addition, the main result of this paper can also be used to show that various non-central distributions (chi-square, "$t$", "$F$") are Schur-concave in the noncentrality parameter.

Article information

Ann. Statist., Volume 2, Number 6 (1974), 1189-1200.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H99: None of the above, but in this section
Secondary: 26A86

Majorization partial orderings probability inequalities exchangeable random variables bounds for distribution functions multivariate normal distribution multivariate $t$ distribution multivariate beta distribution multivariate chi-square distribution associated random variables non-central distributions survival functions


Marshall, Albert W.; Olkin, Ingram. Majorization in Multivariate Distributions. Ann. Statist. 2 (1974), no. 6, 1189--1200. doi:10.1214/aos/1176342873.

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