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.     

Monotonically Overrelaxed EM Algorithms

We explore the idea of overrelaxation for accelerating the expectation-maximization (EM) algorithm, focusing on preserving its simplicity and monotonic convergence properties. It is shown that in many cases, a trivial modification in the M-step results in an algorithm that maintains monotonic increase in the log-likelihood, but can have an appreciably faster convergence rate, especially when EM is very slow. The method is applicable to more general fixed point algorithms. Its simplicity and effectiveness are illustrated with several statistical problems, including probit regression, least absolute deviations regression, Poisson inverse problems, and finite mixtures. This article has supplemental materials available online.     

ML Estimation of the MultivariatetDistribution and the EM Algorithm

Maximum likelihood estimation of the multivariatetdistribution, especially with unknown degrees of freedom, has been an interesting topic in the development of the EM algorithm. After a brief review of the EM algorithm and its application to finding the maximum likelihood estimates of the parameters of thetdistribution, this paper provides new versions of the ECME algorithm for maximum likelihood estimation of the multivariatetdistribution from data with possibly missing values. The results show that the new versions of the ECME algorithm converge faster than the previous procedures. Most important, the idea of this new implementation is quite general and useful for the development of the EM algorithm. Comparisons of different methods based on two datasets are presented.