## The Annals of Statistics

### Majorization in Multivariate Distributions

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176342873

Digital Object Identifier
doi:10.1214/aos/1176342873

Mathematical Reviews number (MathSciNet)
MR362705

Zentralblatt MATH identifier
0292.62037

JSTOR