Exact Inference with Monotone Incomplete Multivariate Normal Data

We consider problems in finite-sample inference with two-step, monotone incomplete data drawn from Nd(μ,Σ), a multivariate normal population with mean μ and covariance matrix Σ. We derive stochastic representations for the distributions of μ̂ and Σ̂, the maximum likelihood estimators of μ and Σ, respectively. 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 Σ̂ and of the estimated regression matrix, Σ̂12Σ̂ 22 . We obtain a representation for Σ̂ in terms of independent matrices, and then derive the exact density function and saddlepoint approximations thereof for Σ̂ and its partial Iwasawa coordinates. We obtain ellipsoidal confidence regions for μ based on T , a generalization of Hotelling’s T -statistic; we derive probability inequalities for, and the asymptotic distribution of, T 2 under various assumptions on the sizes of the complete and incomplete samples, and we apply these results to construct confidence regions for linear combinations of μ and ellipsoidal prediction regions for future complete observations from Nd(μ,Σ). We develop simultaneous confidence intervals for the components of μ, derive upper and lower bounds on the corresponding confidence coefficient, and establish an upper bound for the L∞ distance between the density function of μ̂ and a normal approximation. We also study several testing problems on μ and Σ. For the null hypothesis H0 : Σ = Σ0 where Σ0 is a given matrix, we establish unbiasedness of a modification to the likelihood ratio criterion and obtain its null and non-null distributions. In testing H0 : (μ,Σ) = (μ0,Σ0) where μ0 and Σ0 both 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, but the issue of unbiasedness remains open. In testing H0 : Σ12 = 0 we show that a modified locally most powerful invariant statistic has the same distribution as that of a Bartlett-Pillai-Nanda trace statistic in multivariate analysis of variance. ∗Washington Department of Fish and Wildlife, Olympia, WA 98501, USA. †Department of Statistics, Penn State University, University Park, PA 16802; and The Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, NC 27709, USA. †Supported in part by National Science Foundation grants AST-0434234 and DMS-0112069. 2000 Mathematics Subject Classification: Primary 62H10; Secondary 60D10, 62E15.