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 Σˆ and of the estimated regression matrix, Σˆ12Σˆ22−1. We represent Σˆ 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 Σˆ 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∝Ip+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.