Power comparisons of tests of two multivariate hypotheses based on four criteria

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Abstract
Tests of two hypotheses are considered in this paper, namely, (i) the independence between ap-set and a q-set of variates in a (p + q)-variate normal population, and (ii) the equality of mean vectors of lp-variate normal populations having a common covariance matrix, based on the following four test criteria: (1) Roy's largest root criterion, (2) U(p), a constant times Hotelling's T02: criterion, (3) Pillai's V(p) criterion and (4) Wilks's criterion. For p = 2, the exact non-central C.D.F. is obtained for each of the criteria (2) to (4). For criterion (1) such a study has been made earlier by Pillai, but in this paper approximations to the C.D.F. of the largest root have been obtained in the linear case for p = 2, 3 and 4. Further, for p = 2, extensive power function tabulation has been carried out for criteria (2)−(4) for tests of both (i) and (ii). Comparisons of their power functions have been made. Pillai's V(2) criterion has optimum local properties; for small deviations from the null hypothesis V(2) shows larger power. The power of the largest root criterion is below those of the other three criteria.

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