Suppose X = (X1,...,Xk) T is multivariate normal with unknown mean vector (1,...,k) T and known nonsingular

Suppose X = (X1,...,Xk)

T is multivariate normal with unknown mean vector (θ1,...,θk)

T and known nonsingular covariance matrix Σ.

Consider testing the null hypothesis θi = 0 for all i against θi = 0 for some i. Let C be any closed convex subset of k-dimensional Euclidean space, and let φ be the test that accepts the null hypothesis if X falls in C. Show that φ is admissible.

Hint: First assume Σ is the identity and use Theorem 6.7.1. [An alternative proof is provided by Strasser (1985, Theorem 30.4).]