A note on the power superiority of the restricted likelihood ratio test

Let be a closed convex cone which contains a linear subspace . We investigate the restricted likelihood ratio test for the null and alternative hypotheses based on an n-dimensional, normally distributed random vector (X1,...,Xn) with unknown mean and known covariance matrix [Sigma]. We prove that if the true mean vector satisfies the alternative hypothesis HA, then the restricted likelihood ratio test is more powerful than the unrestricted test with larger alternative hypothesis [real]n. The proof uses isoperimetric inequalities for the uniform distribution on the n-dimensional sphere and for n-dimensional standard Gaussian measure.Order restricted inference Convex cone Gaussian isoperimetric inequality