Uniformly more powerful tests for hypotheses about linear inequalities when the variance is unknown

Let X be a $p$-dimensional normal random vector with unknown mean $\mu$ and covariance matrix $\Sigma =\sigma ^{2}\Sigma _{0}$, where $\Sigma _{0}$ is a known matrix and $\sigma ^{2}$ an unknown parameter. This paper gives a test for the null hypothesis that $\mu$ lies either on the boundary or in the exterior of a closed, convex polyhedral cone versus the alternative hypothesis that $\mu$ lies in the interior of the cone. Our test is uniformly more powerful than the likelihood ratio test.