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

We consider problems in finite-sample inference with two-step, monotone incomplete data drawn from N_d (μ, Σ), a multivariate normal population with mean μ and covariance matrix Σ. We derive a stochastic representation for the exact distribution of over(μ, ̂), the maximum likelihood estimator of μ. We obtain ellipsoidal confidence regions for μ through T², a generalization of Hotelling's statistic. We derive the asymptotic distribution of, and probability inequalities for, T² under various assumptions on the sizes of the complete and incomplete samples. Further, we establish an upper bound for the supremum distance between the probability density functions of over(μ, ̂) and over(μ, ̃), a normal approximation to over(μ, ̂).