On Stochastic Majorization of the Eigenvalues of a Wishart Matrix

In multivariate statistical analysis. orthogonally invariant sets of real positive definite pxp matrices occur as acceptance regions for tests of invariant hypotheses concerning the covariance matrix [ of a multivariate normal distribution. Equivalently. orthogonally invariant acceptance regions can be expressed in terms of the eigenvalues I, (S). .... lp(S) of a random Wishart matrix S Wp(n. [) with n degrees of freedom and expectation nL The probabi I it ies of such regions depend on [ only through A, ([) ..... Ap([). the eigenvalues of L In this paper. the behavior of these probabi I i ties is studied when some Ai increase while others decrease. Our results will be expressed in terms of the tnejorizetion ordering applied to the vector J.l == ()l,([) . ... . )lp([)). where )lp([) = log Ap([L and have implications for the unbiaseoness and monotonicity of the power functions of orthogonally invariant tests.