In a variety of standard multivariate normal testing problems, it is shown that certain procedures, often fully invariant, similar, and/or likelihood ratio, are admissible Bayes procedures. The problems include the multivariate general linear hypothesis (where some of the procedures considered were previously shown to be admissible by other methods), the testing of independence of sets of variates (where the likelihood ratio test is shown, for the first time, to be admissible), tests about only some components of the means, classification procedures (for any number of populations), Behrens-Fisher problem, tests about values of or proportionality or equality of covariance matrices, etc. A general technique is developed for obtaining certain Bayes procedures for such problems from the corresponding Bayes procedures relative to a priori distributions of a certain type for problems where nuisance parameter means have been deleted.
"Admissible Bayes Character of $T^2-, R^2-$, and Other Fully Invariant Tests for Classical Multivariate Normal Problems." Ann. Math. Statist. 36 (3) 747 - 770, June, 1965. https://doi.org/10.1214/aoms/1177700051