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

Let C⊂ℜnC⊂ℜn be a closed convex cone which contains a linear subspace LL. We investigate the restricted likelihood ratio test for the null and alternative hypotheses H0:μ¯∈L,HA:μ¯∈C/L based on an nn-dimensional, normally distributed random vector (X1,⋯,Xn)(X1,⋯,Xn) with unknown mean μ¯=(μ1,…,μn) and known covariance matrix ΣΣ. We prove that if the true mean vector μ¯ satisfies the alternative hypothesis HAHA, then the restricted likelihood ratio test is more powerful than the unrestricted test with larger alternative hypothesis ℜnℜn. The proof uses isoperimetric inequalities for the uniform distribution on the nn-dimensional sphere and for nn-dimensional standard Gaussian measure.