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

Abstract

SUMMARY. In this paper, the authors derived asymptotic joint distribution of the eigenvalues of the multivariate analysis of variance matrix in the noncentral case when the under lying distribution is not necessarily multivariate normal. 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 non central case is useful in studying the power of the tests for the equality 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 (1939), Hsu (1939), and Roy (1939) have independently derived the joint distribution of the eigenvalues of the MANOVA matrix in the central case. Hsu (1941) derived the above distribution in the noncentral case when the sample size tends to infinity and the underlying distribution is multi variate 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. In this paper, we extend the result of Hsu to the case when the underlying distribution is not necessarily multivariate normal.