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Estimating the unconstrained mean and covariance matrix is a popular topic instatistics. However, estimation of the parameters of $N_p(\mu,\Sigma)$ underjoint constraints such as $\Sigma\mu = \mu$ has not received much attention. Itcan be viewed as a multivariate counterpart of the classical estimation problemin the $N(\theta,\theta^2)$ distribution. In addition to the usual inferencechallenges under such non-linear constraints among the parameters (curvedexponential family), one has to deal with the basic requirements of symmetryand positive definiteness when estimating a covariance matrix. We derive thenon-linear likelihood equations for the constrained maximum likelihoodestimator of $(\mu,\Sigma)$ and solve them using iterative methods. Generally,the MLE of covariance matrices computed using iterative methods do not satisfythe constraints. We propose a novel algorithm to modify such (infeasible)estimators or any other (reasonable) estimator. The key step is to re-align themean vector along the eigenvectors of the covariance matrix using the idea ofregression. In using the Lagrangian function for constrained MLE (Aitchison etal. 1958), the Lagrange multiplier entangles with the parameters of interestand presents another computational challenge. We handle this by eitheriterative or explicit calculation of the Lagrange multiplier. The existence andnature of location of the constrained MLE are explored within a data-dependentconvex set using recent results from random matrix theory. A simulation studyillustrates our methodology and shows that the modified estimators performbetter than the initial estimators from the iterative methods.

MLE of Jointly Constrained Mean-Covariance of Multivariate Normal Distributions