On Asymptotic Joint Distribution of the Eigenvalues of the Noncentral Manova Matrix for Nonnormal Populations.

The problem of testing the hypothesis of the inequality of the mean vectors of several multivariate populations with a common covariance matrix received considerable attention in the literature. The test procedures are based upon certain functions of the eigenvalues of the multivariate analysis of variance MANOVA matrix. In the univariate case, the MANOVA matrix reduces to the ratio of the between group and within group sums of squares. The joint distribution of the eigenvalues of the MANOVA matrix in the noncentral case is useful in studying the power of the tests for the inequality of the mean vectors. This distribution is also useful in the problems connected with selection of important discriminant functions in the area of classification. Fisher, Hsu, and Roy have independently derived the joint distribution of the eigenvalues of the MANOVA matrix in the central case. Hsu derived the above distribution in the noncentral case when the sample size tends to infinity and the underlying distribution is multivariate normal. In proving the above result, Hsu assumed that the ratios of the sample sizes of the groups to the total sample size tend to constants in the limiting case. This paper extends the result of Hsu to the case when the underlying distribution is not necessarily multivariate normal. Additional keywords include Nonnormal populations.