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

We continue our recent work on finite-sample, i.e., non-asymptotic, inference with two-step, monotone incomplete data from Nd(μ,Σ), 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 b Σ and of the estimated regression matrix, b Σ12 b Σ −1 22 . We obtain a representation for b Σ in terms of independent matrices, and derive its exact density function, thereby generalizing the Wishart distribution to the setting of monotone incomplete data, and obtain saddlepoint approximations for the distributions of b Σ and its partial Iwasawa coordinates. We establish 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 that of a Bartlett-PillaiNanda trace-statistic in multivariate analysis of variance.