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 μ ̂ , the maximum likelihood estimator of μ . We obtain ellipsoidal confidence regions for μ through T 2 , a generalization of Hotelling’s statistic. We derive the asymptotic distribution of, and probability inequalities for, T 2 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 μ ̂ and μ ˜ , a normal approximation to μ ̂ .