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 a complete monotone sample have closed-form expressions (Anderson, 1957; Little and Rubin, 1987) and that the MLEs from incomplete data of a general missing-data pattern can be obtained using the EM algorithm (Dempster, Laird, and Rubin, 1977). For the routine use of algorithms, it is useful to design the most efficient EM implementation, especially for massive data with missing values, which has been one of the current trends of the study of the EM algorithms (Meng and van Dyk, 1997; Liu, 1997a; Liu, Rubin, Wu, 1997). This article gives nice closed-form expressions, analogous to the extension of the Bartlett decomposition (Liu, 1993), for both the MLEs of the parameters and the correspondent Fisher information matrix from incomplete data of monotone missing-data pattern. We give the statistical interpretation of the closed-form expression of the ML estimates in terms of linear regression, which clarifies the relationship between the Bartlett decomposition approach and the factored likelihood method (Little, and Rubin, 1987).