Suppose (X1,...,Xp) have the multivariate normal density (7.51), so that E(Xi) = i and A1 is the known positive definite covariance matrix. The vector of

Suppose (X1,...,Xp) have the multivariate normal density

(7.51), so that E(Xi) = ξi and A−1 is the known positive definite covariance matrix. The vector of means ξ = (ξ1,...,ξp) is known to lie in a given s-dimensional linear space ΠΩ with s ≤ p; the hypothesis to be tested is that ξ lies in a given

(s − r)-dimensional linear subspace Πω of ΠΩ(r ≤ s).

(i) Determine the UMPI test under a suitable group of transformations as explicitly as possible. Find an expression for the power function.

(ii) Specialize to the case of a simple null hypothesis.