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 mu and covariance matrix Sigma. Under the assumption that Sigma is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks of (Sigma) over cap and of the estimated regression matrix, (Sigma) over cap (12) (Sigma) over cap (-1)(22). We represent (Sigma) over cap 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 (Sigma) over cap and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing H-0 : Sigma = Sigma(0), where Sigma(0) is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing H-0 : (mu, Sigma) = (mu(0), Sigma(0)), where it, and Sigma(0) are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, H-0 : Sigma proportional to Ip+q, we obtain the null distribution of the likelihood ratio criterion. In testing H-0 : Sigma(12) = 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. (C) 2009 Elsevier Inc. All rights reserved.