Asymptotic Representations of the Densities of Canonical Correlations and Latent Roots in MANOVA When the Population Parameters Have Arbitrary Multiplicity

Abstract

Asymptotic representations of the joint densities of the canonical correlation coefficients, calculated from a sample from a multivariate normal population, and of the latent roots of $B(B + W)^{-1},$ where $B$ is $W_p(n_1, \Sigma, \Omega)$ and $W$ is $W_p(n_2, \Sigma),$ are obtained by deriving asymptotic representations of the hypergeometric functions in the joint densities. The results hold in the first case for large sample size and arbitrary values of the population canonical correlations and in the second case for large $n_2$ and $\Omega = n_2\Theta,$ where the latent roots of the noncentrality matrix $\Omega$ are arbitrary.