A test of a multivariate normal mean with composite hypotheses determined by linear inequalities

In this paper we propose a new multivariate generalization of a one-sided test in a way-different from that of Kud{circled ring operator} (1963). Let X be a p-variate normal random variable with the mean vector μ. and a known covariance matrix. We consider the null hypothesis that μ. lies on the boundary of a convex polyhedral cone determined by linear inequalities; the alternative is that μ lies in its interior. A two-sided version is also discussed. This paper provides likelihood ratio tests and some applications along with some discussion of the geometry of convex polyhedral cones.