MAXIMUM LIKELIHOOD ESTIMATION FOR LINEAR GAUSSIAN COVARIANCE MODELS

We study parameter estimation in linear Gaussian covariance models, which are p-dimensional Gaussian models with linear con-straints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex optimization problem which typically has many local optima. We prove that the log-likelihood function is concave over a large region of the cone of positive definite matrices. Using recent results on the asymptotic distribution of ex-treme eigenvalues of the Wishart distribution, we provide sufficient conditions for any hill climbing method to converge to the global op-timum. The proofs of these results utilize large-sample asymptotic theory under the scheme n/p → γ> 1. Remarkably, our numeri-cal simulations indicate that our results remain valid for min{n, p} as small as 2. An important consequence of this analysis is that for sample sizes n ' 14p, maximum likelihood estimation for linear Gaus-sian covariance models behaves as if it were a convex optimization problem. 1. Introduction. I