The General Canonical Correlation Distribution

Abstract

The general canonical correlation distribution is given as a multiple power series in the true canonical correlations $\rho_i$. When only one true correlation is not zero, this series is expressible as a generalized hypergeometric function, for the cases both of non-central means and of correlations proper. In the general case of more than one non-zero true correlation the coefficients in the expansion depend on the conditional moments of the sample correlations between the pairs of transformed variables representing the true canonical variables, when the sample canonical correlations between the sample canonical variables are fixed. Methods are given of obtaining these coefficients for both cases, non-central means and correlations proper; and their form up to the fourth order, corresponding to $O(\rho^8)$ in the expansion, listed in Appendix I. The detailed terms making up these coefficients are given, in the case of two non-zero correlations, up to the fourth order, and in the general case, up to the third order, in Appendix II.