The problem of testing the general multivariate linear hypothesis, also known as MANOVA, and the problem of testing independence between two sets of multivariate normal random variables are the two problems considered in this paper. In  two sufficient conditions for the power function of an invariant test of the general linear hypothesis to be a monotone increasing function of each of the noncentrality parameters have been obtained. In  one of these two conditions has been extended to invariant tests of the hypothesis of independence between two sets of variates. These conditions are in terms of, respectively, the convexity and the symmetry of certain sections of the acceptance regions of the tests; and their verification is, in general, nontrivial. In this paper it is shown that the power functions of the members of a class of invariant tests based on statistics "generated" by symmetric gauge functions of convex increasing functions of the maximal invariants are monotone increasing functions of the relevant noncentrality parameters. In this process we have explained an interesting tie-up between the monotonicity properties of the invariant tests for the two problems (Theorem 2) and have obtained extension of some results on the symmetric gauge functions and convexity in the matrix theory (Theorem 3 and Theorem 4). The terms "invariance," "invariant tests," and "maximal invariants" have been used throughout this paper in connection with the relevant groups of transformations mentioned in the Section 2 without their explicit mention each time.
"A Class of Tests with Monotone Power Functions for Two Problems in Multivariate Statistical Analysis." Ann. Math. Statist. 36 (6) 1794 - 1801, December, 1965. https://doi.org/10.1214/aoms/1177699808