Optimum invariant tests for random manova models

Consider the canonical‐form MANOVA setup with X: n × p = (+ E, X i n i × p, i = 1, 2, 3, M i : n i × p, i = 1, 2, n 1 + n 2 + n 3 ) p , where E is a normally distributed error matrix with mean zero and dispersion I n (> 0 (positive definite). Assume (in contrast with the usual case) that M 1 i is normal with mean zero and dispersion I n 1) and M 2 2 is either fixed or random normal with mean zero and different dispersion matrix I n 2 (being unknown. It is also assumed that M 1 E , and M 2 (if random) are all independent. For testing H 0 ) = 0 versus H 1 : (> 0, it is shown that when either n 2 = 0 or M 2 is fixed if n 2 > 0, the trace test of Pillai (1955) is uniformly most powerful invariant (UMPI) if min( n 1 , p )= 1 and locally best invariant (LBI) if min( n 1 p ) > 1 underthe action of the full linear group Gl ( p ). When p > 1, the LBI test is also derived under a somewhat smaller group G T (p) of p × p lower triangular matrices with positive diagonal elements. However, such results do not hold if n 2 > 0 and M 2 is random. The null, nonnull, and optimality robustness of Pillai's trace test under Gl( p ) for suitable deviations from normality is pointed out.