Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data

It is well known that the maximum likelihood estimates (MLEs) of a multivariate normal distribution from incomplete data with a monotone pattern have closed-form expressions and that the MLEs from incomplete data with a general missing-data pattern can be obtained using the Expectation-Maximization (EM) algorithm. This article gives closed-form expressions, analogous to the extension of the Bartlett decomposition, for both the MLEs of the parameters and the associated Fisher information matrix from incomplete data with a monotone missing-data pattern. For MLEs of the parameters from incomplete data with a general missing-data pattern, we implement EM and Expectation-Constrained-Maximization-Either (ECME), by augmenting the observed data into a complete monotone sample. We also provide a numerical example, which shows that the monotone EM (MEM) and monotone ECME (MECME) algorithms converge much faster than the EM algorithm.