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

Let C ⊂ ℜ n be a closed convex cone which contains a linear subspace L . We investigate the restricted likelihood ratio test for the null and alternative hypotheses H 0 : μ ¯ ∈ L , H A : μ ¯ ∈ C / L based on an n -dimensional, normally distributed random vector ( X 1 , ⋯ , X n ) with unknown mean μ ¯ = ( μ 1 , … , μ n ) and known covariance matrix Σ . We prove that if the true mean vector μ ¯ satisfies the alternative hypothesis H A , then the restricted likelihood ratio test is more powerful than the unrestricted test with larger alternative hypothesis ℜ n . The proof uses isoperimetric inequalities for the uniform distribution on the n -dimensional sphere and for n -dimensional standard Gaussian measure.