Simultaneous Tests for the Equality of Covariance Matrices Against Certain Alternatives

Abstract

In many situations, it is of interest to test for the equality of variances or covariance matrices against certain alternatives. Hartley [6] considered the problem of testing for the equality of variances against the alternative that at least one variance is different from the other. Gnanadesikan [3] considered the problem of testing for the equality of variances against the alternative that at least one variance is not equal to the standard. Recently, Krishnaiah [12] considered testing for the equality of variances against the alternative that at least one variance is not equal to the next. In the above procedures, it was assumed that the underlying populations are univariate normal. In this paper, we consider multivariate generalizations of the above test procedures. The procedures proposed in this paper are based upon expressing the total hypothesis as the intersection of some elementary hypotheses and testing these elementary hypotheses by using conditional distributions. In the two sample case, our procedures are similar (but not equivalent) to the procedure proposed by Roy [16]; the test statistics used by him in testing some of the elementary hypotheses are different from those used in this paper.