Finite-sample inference with monotone incomplete multivariate normal data, II

We continue our recent work on inference with two-step, monotone incomplete data from a multivariate normal population with mean μ and covariance matrix Σ. Under the assumption that Σ is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks of over(Σ, ̂) and of the estimated regression matrix, over(Σ, ̂)₁₂ over(Σ, ̂)₂₂^{- 1}. We represent over(Σ, ̂) in terms of independent matrices; derive its exact distribution, thereby generalizing the Wishart distribution to the setting of monotone incomplete data; and obtain saddlepoint approximations for the distributions of over(Σ, ̂) and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing H₀ : Σ = Σ₀, where Σ₀ is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing H₀ : (μ, Σ) = (μ₀, Σ₀), where μ₀ and Σ₀ are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, H₀ : Σ ∝ I_{p + q}, we obtain the null distribution of the likelihood ratio criterion. In testing H₀ : Σ₁₂ = 0 we show that a modified locally most powerful invariant statistic has the same distribution as a Bartlett-Pillai-Nanda trace statistic in multivariate analysis of variance.