On stochastic majorization of the eigenvalues of a Wishart matrix Original Research Article

In multivariate statistical analysis, orthogonally invariant sets of real positive definite p × p matrices occur as acceptance regions for tests of invariant hypotheses concerning the covariance matrix Σ of a multivariate normal distribution. Equivalently, orthogonally invariant acceptance regions can be expressed in terms of the eigenvalues l1(S), …, lp(S) of a random Wishart matrix S not, vert, similar Wp(n, Σ) with n degrees of freedom and expectation nΣ. The probabilities of such regions depend on Σ only though λ1(Σ), …, λp(Σ), the eigenvalues of Σ. In this paper, the behavior of these probabilities is studied when some λi increase while others decrease. Our results will be expressed in terms of the majorization ordering applied to the vector μ ≡ (μ1(Σ), …, μp(Σ)), where μi(Σ) = log λi(Σ), and have implications for the unbiasedness and monotonicity of the power functions of orthogonally invariant tests.