The noncentral distribution of latent roots arising in several situations in multivariate analysis involves the integration of a hypergeometric function of matrix variates over a group of orthogonal matrices in the real case and that of unitary matrices in the complex case. In this paper the subgroup of the orthogonal group (unitary group) for which the integrand is maximized has been found under mild restrictions. The results of earlier authors (Anderson, Chang, James, Li and Pillai) follow as special cases. Further, the maximization results concerning the integrand have been used to study asymptotic expansions of the distributions of the characteristic roots of matrices arising in canonical correlation analysis and MANOVA when the corresponding parameter matrices have several multiple roots.
"Maximization of an Integral of a Matrix Function and Asymptotic Expansions of Distributions of Latent Roots of Two Matrices." Ann. Statist. 4 (4) 796 - 806, July, 1976. https://doi.org/10.1214/aos/1176343554