It is an open problem to construct a test for structural relationship among the mean vectors of several multivariate normal populations with unequal covariance matrices. In this paper some solutions to this problem are provided when the unequal covariance matrices are either completely known or possess a special structure and are partially known. A class of admissible Bayes tests is characterized when the covariance matrices are completely known. When the covariance matrices [Sigma]1,...,[Sigma]k have the structure [Sigma]i = [sigma]2iV, where [sigma]2i are positive constants and V is a positive definite matrix, we have derived some meaningful tests using the union-intersection principle in conjunction with the Hotelling's T2-test under the assumption that either [sigma]2i's are known or V is known. We have also provided an alternative derivation of the likelihood ratio test for the case of equal covariance matrices using the union-intersection principle, and in the sequel proposed a few new tests.
