The Distribution of the Characteristic Roots of $S_1S_2^{-1}$ Under Violations

The paper deals with the density of the characteristic roots of $\mathbf{S}_1\mathbf{S}_2^{-1}$ where $\mathbf{S}_1$ has a noncentral Wishart distribution, $W(p, n_1, \mathbf{\Sigma}_1, \mathbf{\Omega})$, and $\mathbf{S}_2$ has an independently distributed central Wishart distribution $W(p, n_2 \mathbf{\Sigma}_2, \mathbf{0})$, under a condition. This density is basic for an exact study of robustness of tests of at least two multivariate hypotheses.