Locally minimax tests for a multinormal data problem

LetX be ap-normal random vector with unknown mean μ and unknown covariance matrix Σ and letX be partitioned asX=(X (1) ′ ,X (2) ′ , ...,X (r) ′ )′ whereX(j) is a subvector of dimensionpj such that ∑ j=1 r p j =p. We show that the tests, obtained by Dahel (1988), are locally minimax. These tests have been derived to confront Ho: μ=0 versusH1: μ≠0 on the basis of sample of sizeN, X1, ..., XN, drawn fromX andr additional samples of sizeNj, U i (j) , i=1, ..., Nj, drawn fromX(1), ...X(r) respectively. We assume that the (r+1) samples are independent and thatNj>pj forj=0, 1, ..., r (No≡N andpo≡p). Whenr=2 andp=2, a Monte Carlo study is performed to compare these tests with the likelihood ratio test (LRT) given by Srivastava (1985). We also show that no locally most powerful invariant test exists for this problem.