The exact noncentral distribution of the generalized multiple correlation matrix

Given p×n X N(βY, ∑?I), β, ∑ unknown, the noncentral multivariate beta density of the matrix L = [(YY′)^-1/2Y X′ (XX′)^-1XY′ (YY′)^-1/2] is desired. Khatri (1964) finds this density when β is of rank unity. The present paper derives the noncentral density of L and the density of the roots matrix of L for full rank β. The dual case density of L is also obtained. The derivations are based on generalized Sverdrup's lemma, Kabe (1965), and the relationship between primal and dual density of L is explicitly established.