The Ordering of Shannon Entropies for the Multivariate Distributions and Distributions of Eigenvalues

In this paper, we prove the Shannon entropy inequalities for the multivariate distributions via the notion of convex ordering of two multivariate distributions. We further characterize the multivariate totally positive of order 2 ([Formula: see text]) property of the distribution functions of eigenvalues of both central Wishart and central MANOVA models, and of both noncentral Wishart and noncentral MANOVA models under the general population covariance matrix set-up. These results can be directly applied to both the comparisons of two Shannon entropy measures and the power monotonicity problem for the MANOVA problem.