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 StatisticsOpen Access

Majorization in Multivariate Distributions

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Reveals Schur-concavity in multivariate distributions, highlighting implications for density functions and inequalities.

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.

Marshall et al. (1974) studied this question.

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