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analytic function arranged in nonincreasing AS2A ASgA ASjA Bahadur efficiency belongs cd.f component test matrix consider covariance matrix decreasing values defined denoted diagonal elements diagonal matrix elementary symmetric functions equal exp(X H is true hypothesis H i-th IDENTITY MATRIX intersection lemma Let f(x likelihood ratio component likelihood ratio test limiting distribution linear model log(l+X matrix see Anderson matrix whose diagonal max tr maximizes maximum monotonicity property multivariate tests nonincreasing order nonsingular matrix nonzero characteristic roots normal distribution null component hypotheses null hypothesis obtain ordered characteristic roots orthogonal matrix p-k+m+1 partition power function ratio component test reject H rejection region respectively roots of ASA row-orthogonal sets of variates simultaneous confidence bounds step-down procedure Suppose symmetric matrix teristic roots test based test for H test statistic Theorem 2.1 tr(ASA union-intersection principle union-intersection test values of f(9 wish to test Wishart distribution