The problem of maximum likelihood estimation of Lowner ordered covariance matrices is considered. It is shown that a dual formulation of this problem is tractable and important in its own right. The interplay between the primal and dual problems suggests a general algorithm for computing the solutions to these problems. This algorithm has application to some estimation problems in balanced multivariate variance components models. The speed of convergence is also discussed for the variance components models.
"Maximum Likelihood Estimation of a Set of Covariance Matrices Under Lowner Order Restrictions with Applications to Balanced Multivariate Variance Components Models." Ann. Statist. 19 (2) 850 - 869, June, 1991. https://doi.org/10.1214/aos/1176348124