In this paper, the exact distribution of Wilks' likelihood ratio criterion, $\Lambda$, for MANOVA in the noncentral linear case i.e. when the alternative hypothesis is of unit rank, has been obtained and explicit expressions for the same for $p = 2, 3, 4$ and 5, where $p$ is the number of variables and for general $f_1$ and $f_2$ are given. A general form of the distribution of $\Lambda$ in this case, for any $p$, is also given. It has been shown that the total integral of the series obtained by taking a few terms only, rapidly approaches the theoretical value one as more terms are taken into account. Further the accuracy of the approximation, suggested by Posten and Bargmann , is examined numerically and it has been shown that the approximation is excellent except when $f_1$ and $f_2$ are both small and the noncentrality parameter $\lambda^2$ is large.
A. K. Gupta. "Noncentral Distribution of Wilks' Statistic in Manova." Ann. Math. Statist. 42 (4) 1254 - 1261, August, 1971. https://doi.org/10.1214/aoms/1177693238