Schur^2-concavity properties of Gaussian measures, with applications to hypotheses testing

TL;DR

This paper explores how Gaussian measures exhibit Schur-concavity and Schur-convexity properties, with implications for hypothesis testing in multivariate analysis, highlighting conditions under which these properties are strict or hold.

Contribution

It establishes new Schur-concavity and Schur-convexity properties of Gaussian measures related to set symmetry, with applications to hypothesis testing on multivariate means.

Findings

Probability P(Z ∈ A + θ) is Schur-concave/convex in squared parameters.

Strictness of properties unless set A is spherically symmetric.

Applications to hypothesis testing on multivariate means.

Abstract

The main results imply that the probability P(\ZZ\in A+\th) is Schur-concave/Schur-convex in (\th_1^2,\dots,\th_k^2) provided that the indicator function of a set A in \R^k is so, respectively; here, \th=(\th_1,\dots,\th_k) in \R^k and \ZZ is a standard normal random vector in \R^k. Moreover, it is shown that the Schur-concavity/Schur-convexity is strict unless the set A is equivalent to a spherically symmetric set. Applications to testing hypotheses on multivariate means are given.