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 ( θ 1 2 , … , θ k 2 ) provided that the indicator function of a set A in R k is so, respectively; here, θ = ( θ 1 , … , θ k ) ∈ R k and Z 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.