Comparison of powers of a class of tests for multivariate linear hypothesis and independence

In this paper we consider the class C of test statistics T = [Sigma]j = 1p Q(lj) for testing the multivarite linear hypothesis and independence of two sets of variables. Here Q(l) is a monotone increasing function for l >= 0, Q(0) = 0, Q'(0) = 1, and has a continuous third order derivative in a neighborhood of l = 0. This class includes the likelihood ratio test, Lawley-Hotelling trace test and Bartlett-Nanda-Pillai trace test. We compare the local powers of tests on the basis of the asymptotic expansions of their distributions. The differences in the powers of all the test in C can be explained in terms of [gamma] = Q''(0). The comparisons reveal no uniform superiority properties.