Necessary and sufficient conditions for an invariant test to be admissible among invariant tests in the general multivariate analysis of variance problem are presented. It is shown that in many cases the popular tests based on the likelihood ratio matrix are inadmissible. Other tests are shown admissible. Numerical work suggests that the inadmissibility of the likelihood ratio test is not serious. The results are given for the multivariate analysis of variance problem as a special case.
"Admissibility of Invariant Tests in the General Multivariate Analysis of Variance Problem." Ann. Statist. 11 (4) 1086 - 1099, December, 1983. https://doi.org/10.1214/aos/1176346323