Asymptotic expansion of the null distribution of LR statistic for multivariate linear hypothesis when the dimension is large

In this paper we derive an asymptotic expansion of the null distribution of likelihood ratio statistic for multivariate linear hypothesis when the dimension is comparable to the sample size. Our asymptotic approximations are numerically compared with some other approximations including the large sample approximation due to Box [Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika 36:317-346]. It is shown that the approximations proposed in this paper are good when the dimension is large and close to the sample size, and further our approximations are similar to the Box's approximation even for most of the usual large sample cases.