srOCHASTIC MAJORIZATION OF THE LOG-EIGENVALUES OF A BIVARIATE WISHART MATRIX1.2

Let 1 = (l1 ' l2) and A = (A1. A2). where A1;:f; A2> 0 are the ordered eigenvalues of Sand:E, respectively, and S.... Wz(n, E) is a bivariate Wishart matrix. Let m = (mlo m2) and JL = (f.L1. f.L2). where 1T'/i =log 4 and /-4. =log Ai ' It is shown that P _fm jl ' B l is Schur-convex in J,l whenever B is a Schur-monotone set, i.e. [x E B, x majorizes x"]::;> x · E B. This result implies the unbiased-ness and power-monotonicity of a class of invariant tests for bivari-ate sphericity and other orthogonally invariant hypotheses