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, n₁ + n₂ + n₃) 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 M₁i is normal with mean zero and dispersion I_n1) and M₂2 is either fixed or random normal with mean zero and different dispersion matrix I_n2 (being unknown. It is also assumed that M₁ E, and M₂ (if random) are all independent. For testing H₀) = 0 versus H₁: (> 0, it is shown that when either n₂ = 0 or M₂ is fixed if n₂ > 0, the trace test of Pillai (1955) is uniformly most powerful invariant (UMPI) if min(n₁, p)= 1 and locally best invariant (LBI) if min(n₁ 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 n₂ > 0 and M₂ is random. The null, nonnull, and optimality robustness of Pillai's trace test under Gl(p) for suitable deviations from normality is pointed out.