Maximum likelihood estimators are obtained for multivariate components of variance models under the condition that the effect covariance matrix is positive semidefinite with a maximum rank. The rank of the estimator is random. The estimation procedure leads to a likelihood ratio test that the rank of the effect matrix is not greater than a given number against the alternative that the rank is not greater than a larger specified number. Linear structural relationship models and some factor analytic models can be put into this framework.
"Maximum Likelihood Estimators and Likelihood Ratio Criteria in Multivariate Components of Variance." Ann. Statist. 14 (2) 405 - 417, June, 1986. https://doi.org/10.1214/aos/1176349929