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 Σ ˆ and of the estimated regression matrix, Σ ˆ 12 Σ ˆ 22 − 1 . We represent Σ ˆ 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 Σ ˆ and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing H 0 : Σ = Σ 0 , where Σ 0 is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing H 0 : ( μ , Σ ) = ( μ 0 , Σ 0 ) , where μ 0 and Σ 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 : Σ ∝ I p + q , we obtain the null distribution of the likelihood ratio criterion. In testing H 0 : Σ 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.