Lower bounds for the distributions of certain multivariate test statistics

A lower (upper) bound is given for the distribution of each d_j, j = k + 1, …, p (j = 1, …, s), the jth latent root of AB^?1, where A and B are independent noncentral and central Wishart matrices having $\text{W}{}_{\mathrm{p}}\text{(q, Σ; Ω)}$ with rank $\text{(Ω) ≤ k = p ? s}$ and W_p(n, Σ), respectively. Similar bound are also given for the distributions of noncentral means and canonical correlations. The results are applied to obtain lower bounds for the null distributions of some multivariate test statistics in Tintner's model, MANOVA and canonical analysis.