Exact joint distributions of any few ordered roots of a class of random matrices

In this paper, the authors derived exact expressions for the joint marginal densities of any few consecutive ordered roots of a class of random matrices which includes Wishart matrix, MANOVA matrix, and canonical correlation matrix. In addition, the authors obtained an exact expression for the probability integral associated with the joint distribution of any two ordered roots as well as an expression for the c.d.f. of a single intermediate root. All the above expressions are linear combinations of the products of certain double integrals which can be evaluated without any difficulty.