On the Marginal Distributions of the Latent Roots of the Multivariate Beta Matrix

Abstract

On the marginal distributions of the latent roots of the multivariate beta matrix The marginal distributions of the latent roots of the multivariate beta matrix are shown to constitute a complete system of solutions of an ordinary differential equation (d.e.), which is related to the author's d.e. 's for Rotelling's generalized T 2 and Pillai's V(m) statistics. Similar results are o given for the latent roots of the multivariate F and Wishart matrices (E=I). Pillai's approximations to the distributions of the largest and smallest roots are interpreted as exact solutions, the contributions of higher order solutions being neglected. m ~(q-m-l) m ~(n-m-l) The marginal distributions of the individual £, have been investigated by ~ (1) (2) On the marginal distributions of the latent roots of the multivariate beta mattix* q,n ~ m. The latent roots £1 > • • • > £m > 0 of the multivariate beta matrix B = S(S+T)-l are well known to have the joint density function