Lattice-ordered conditional independence models for missing data

Statistical inference for the parameters of a multivariate normal distribution Np([mu], [Sigma]) based on a sample with missing observations is straightforward when the missing data pattern is monotone (= nested), reducing to the analysis of several normal linear regression models by step-wise conditioning. When the missing data pattern is non-monotone, however, such analysis is impossible. It is shown here that every missing data pattern naturally determines a set of lattice-ordered conditional independence restrictions which, when imposed upon the unknown covariance matrix [Sigma], yields a factorization of the joint likelihood function as a product of (conditional) likelihood functions of normal linear regression models just as in the monotone case. From this factorization the maximum likelihood estimators of [mu] and [Sigma] (under the conditional independence restrictions) can be explicitly derived.