Schur2-concavity properties of Gaussian measures, with applications to hypotheses testing

The main results imply that the probability P(Z∈A+θ) is Schur-concave/Schur-convex in (θ12,…,θk2) provided that the indicator function of a set A in Rk is so, respectively; here, θ=(θ1,…,θk)∈Rk and Z is a standard normal random vector in Rk. 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.