MAXIMUM LIKELIHOOD COVARIANCE ESTIMATION WITH A CONDITION NUMBER CONSTRAINT By

High dimensional covariance estimation is known to be a difficult problem, has many applications and is of current interest to the larger statistical community. We consider the problem of estimating the covariance matrix of a multivariate normal distribution in the “large p small n” setting. Several approaches to high dimensional covariance estimation have been proposed in the literature. In many applications, the covariance matrix estimate is required to be not only invertible but also well-conditioned. Although many estimators attempt to do this, none of them address this problem directly. In this paper, we propose a maximum likelihood approach with an explicit constraint on the condition number to try and obtain a well-conditioned estimator. We demonstrate that the proposed estimation approach is computationally efficient, can be interpreted as a type of nonlinear shrinkage estimator, and has a natural Bayesian interpretation. We fully investigate the theoretical properties of the proposed estimator and proceed to develop an approach that adaptively determines the level of regularization that is required. Finally we investigate the performance of the estimator in simulated and real-life examples and demonstrate that it has good risk properties and can serve as a competitive procedure especially when the sample size is small and when a well-conditioned estimator is required.