Power comparisons of tests of two multivariate hypotheses based on individual characteristic roots

Summary In this paper, power comparisons are made for tests of each of the following two hypotheses based on individual characteristic roots of a matrix arising in each case: (i) independence between ap-set and aq-set of variates in a (p+q)-variate normal population withp≦q and (ii) equality ofp-dimensional mean vectors ofl p-variate normal populations having a common covariance matrix. At first, a few lemmas are given which help to reduce the central distributions of the largest, smallest, second largest, and the second smallest roots in terms of incomplete beta functions or functions of them. Since the central distribution of the largest root has been discussed by Pillai earlier in several papers ([6], [8], [9], [11], [12], [13]) cdf’s of the three others in the central case are given. Further, the non-central distributions of the individual roots forp-3 are considered for the two hypotheses and that of the smaller root forp=2; that of the largest root forp=2 has been obtained by Pillai earlier, (Pillai [11], Pillai and Jayachandran [14]). The work of this author was supported by the National Science Foundation Grant No. GP-4600.