This dissertation considers homogeneity test for comparing multivariate normal populations with generalized order restricted alternative hypothesis. The framework in this dissertation presents a generalized multivariate order for the mean vectors. This order is induced from a closed convex cone in Hilbert space without any speci?cs on particular structures of the cones. Such cones are used to express the generalized order restricted alternative hypothesis. This dissertation derives the restricted maximum likelihood estimators (RMLEs) for the mean vectors under multivariate order restrictions, develops the likelihood ratio tests (LRTs) for the hypotheses about the restricted mean vectors. The statistical procedures are described through the projections onto closed convex cones. These closed convex cones are used to describe both the null and alternative hypotheses. The main result of this work is establishing the unbiasedness of the homogeneity test.