On the distributions of a class of statistics in multivariate analysis

Abstract

The noncentral distributions of Y = Π i=1 p θ i a(1 − θ i) b are obtained, where a and b are known real numbers and θ i 's stand for latent roots of a matrix arising in each of three situations in multivariate normal theory, namely, test of equality of two covariance matrices, MANOVA, and canonical correlation. The study is extended to the complex case as well. The distributions are derived in terms of H-functions as a result of inverse Mellin transforms. Further, asymptotic expansions of the distribution of Y have been obtained in the case of two covariance matrices for selected values of ( a, b).