The MLE algorithm for the matrix normal distribution

The maximum likelihood estimation (MLE) of the parameters of the matrix normal distribution is considered, In the absence of analytical solutions of the system of likelihood equations for the among-row and among-column covariance matrices, a two-stage algorithm must be solved to obtain their maximum likelihood estimators. A necessary and sufficient condition for the existence of maximum likelihood estimators is given and the question of their stability as solutions of the system of likelihood equations is addressed. In particular, the covariance matrix parameters and their maximum likelihood estimators are defined up to a positive multiplicative constant; only their direct product is uniquely defined. Using simulated data under two variance-covariance structures that, otherwise, are indistinguishable by semivariance analysis, further specific aspects of the procedure are studied: (1) the convergence of the MLE algorithm is assessed; (2) the empirical bias of the direct product of covariance matrix estimators is calculated for various sample sizes; and (3) the consistency of the estimator is evaluated by its mean Euclidean distance from the parameter, as a function of the sample size. The adequacy of the matrix normal model, including the separability of the variance-covariance structure, is tested on multiple time series of dental medicine data; other applications to real doubly multivariate data are outlined.