Asymptotic expansions of the null distribution of the MANOVA test statistics including the likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai tests are obtained when both the sample size and the dimension tend to infinity with assuming the ratio of the dimension and the sample size tends to a positive constant smaller than one. Cornish-Fisher expansions of the upper percent points are also obtained. In order to study the accuracy of the approximation formulas, some numerical experiments are done, with comparing to the classical expansions when only the sample size tends to infinity.
"Asymptotic expansions of the distributions of MANOVA test statistics when the dimension is large." Hiroshima Math. J. 44 (3) 247 - 259, November 2014. https://doi.org/10.32917/hmj/1419619745