What question did this study set out to answer?

The aim is to show that certain properties of joint densities being Schur-concave lead to useful implications in multivariate statistics.

November 1, 1974The Annals of Statistics124 citationsOpen Access

Majorization in Multivariate Distributions

AMAlbert W. Marshall IOIngram Olkin

Key Points

The aim is to show that certain properties of joint densities being Schur-concave lead to useful implications in multivariate statistics.
Analyzed joint density functions of multivariate distributions to establish Schur-concavity.
Demonstrated that the convolution of Schur-concave functions is also Schur-concave.
Used results to derive implications for non-central chi-square, t, and F distributions regarding their behavior with respect to the noncentrality parameter.
Found that $P(X \in A + \theta)$ is Schur-concave for Schur-concave indicator functions.
Proved that the convolution of Schur-concave functions maintains the Schur-concavity property.
Established inequalities related to Schur-concave densities in multivariate normal and other distributions.

Abstract

In case the joint density f of X = (X₁, , Xₙ) is Schur-concave (is an order-reversing function for the partial ordering of majorization), it is shown that P (X A +) is a Schur-concave function of 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 X₁, , Xₙ 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.

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