On stochastic majorization of the eigenvalues of a Wishart matrix

In multivariate statistical analysis, orthogonally invariant sets of real positive definite p X p matrices occur as acceptance regions for tests of invariant hypotheses concerning the covariance matrix Sigma of a multivariate normal distribution. Equivalently, orthogonally invariant acceptance regions can be expressed in terms of the eigenvalues l(1)(S),..., l(p)(S) of a random Wishart matrix S similar to W-p(n, Sigma) with n degrees of freedom and expectation n Sigma. The probabilities of such regions depend on Sigma only though lambda(1)(Sigma),..., lambda(p)(Sigma), the eigenvalues of Sigma. In this paper, the behavior of these probabilities is studied when some lambda(i) increase while others decrease. Our results will be expressed in terms of the majorization ordering applied to the vector mu = (mu(1)(Sigma),..., mu(p)(Sigma)), where mu(i)(Sigma) = log lambda(i)(Sigma), and have implications for the unbiasedness and monotonicity of the power functions of orthogonally invariant tests.