Laplace expansion for Bartlett–Nanda–Pillai's test statistic and its error bound

We construct asymptotic expansions for the distribution function of the Bartlett–Nanda–Pillai statistic under the condition that the null linear hypothesis is valid in a multivariate linear model. Computable estimates of the accuracy of approximation are obtained via the Laplace approximation method, which is generalized to integrals for matrix-valued functions.