An Asymptotic Expansion of the Nonnull Distribution of Wilks Criterion for Testing the Multivariate Linear Hypothesis

Abstract

An asymptotic expansion of the nonnull distribution of the Wilks statistic for testing the linear hypothesis in multivariate analysis of variance is obtained up to the order $N^{-2}$ where $N$ is the sample size, for the first time in terms of noncentral beta distributions. The asymptotic distributions are better than the ones available in Anderson (1958) in the null case and in Sugiura and Fujikoshi (1969) and Posten and Bergman (1964) in the nonnull case. In fact, for certain parameters the asymptotic expansion reduces to the first term and we get the exact distribution.