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 constraints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex optimization problem which typically has many local maxima. Using recent results on the asymptotic distribution of extreme eigenvalues of the Wishart distribution, we provide sufficient conditions for any hill climbing method to converge to the global maximum. Although we are primarily interested in the case in which n >> p, the proofs of our results utilize large sample asymptotic theory under the scheme n/ p -> gamma > 1. Remarkably, our numerical simulations indicate that our results remain valid for p as small as 2. An important consequence of this analysis is that, for sample sizes n similar or equal to 14p, maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem.