Maximum likelihood estimation of Gaussian models with missing data - Eight equivalent formulations

In this paper we derive the maximum likelihood problem for missing data from a Gaussian model. We present in total eight different equivalent formulations of the resulting optimization problem, four out of which are nonlinear least squares formulations. Among these formulations are also formulations based on the expectation-maximization algorithm. Expressions for the derivatives needed in order to solve the optimization problems are presented. We also present numerical comparisons for two of the formulations for an ARMAX model.