On the Distribution of a Multiple Correlation Matrix: Non-Central Multivariate Beta Distributions

Abstract

Of several possible versions of multiple correlation matrix between two sets of variables $\mathbf{x}$ and $\mathbf{y}$ (see, e.g., Khatri, 1964), we derive using the techniques of A. T. James (zonal polynomials), the non-null distribution of one version when (i) one of the two sets of variables is fixed, i.e., multivariate analysis of variance and covariance case (MANOVA), and when (ii) both sets of variables are random variables, i.e., canonical correlations case. These distributions are non-central multivariate $\beta$-distributions in much the same way as the two cases of multiple correlation commonly known as the multiple correlation of the second and the first kind respectively.