On computable estimates for accuracy of approximation for the Bartlett–Nanda–Pillai statistic

For the Bartlett-Nanda-Pillai statistic, we find computable estimates for accuracy of approximation, i.e., we describe explicitly the dependence on all parameters of the distributions that occur in the inequalities. For the other two classical statistics traditionally used in multivariate analysis of variance, i.e., the likelihood-ratio and Lawley–Hotelling statistics, similar computable estimates were found earlier.