Tests for the mean of a multivariate normal population

We study the problem of testing: H₀ : μ ∈ P against H₁ : μ ? P, based on a random sample of N observations from a p-dimensional normal distribution N_p(μ, Σ) with Σ > 0 and P a closed convex positively homogeneous set. We develop the likelihood-ratio test (LRT) for this problem. We show that the union-intersection principle leads to a test equivalent to the LRT. It also gives a large class of tests which are shown to be admissible by Stein's theorem (1956). Finally, we give the α-level cutoff points for the LRT.