Jordan Algebras, the EM Algorithm, and Covariance Matrices

Using Jordan algebras, Galois field theory and the EM algorithm we show how to obtain either essentially closed-form or simplified solutions to the ML equations for estimation of the covariance matrix for multivariate normal data in the following situations: (a)Data having a patterned covariance matrix; equivalently, data with a linear covariance structure. This case includes the problems of variance and variance-covariance components estimation; unbalanced repeated measures designs; and some time series models;(b)Data vectors with values missing completely at random; in particular, retrospective analyses of long term, clinical trials, incomplete repeated measures, and time series;(c)The intersection of (a) and (b): multivariate normal data assumed to have a linear covariance structure but with some values missing (completely at random);