Maximum likelihood estimation for linear Gaussian covariance models

Piotr Zwiernik, Caroline Uhler, Donald Richards

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2017 · Volume 79 · Issue 4

Abstract

We study parameter estimation in linear Gaussian covariance models, which arep-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 whichn≫p, the proofs of our results utilize large sample asymptotic theory under the schemen/p→γ>1. Remarkably, our numerical simulations indicate that our results remain valid forpas small as 2. An important consequence of this analysis is that, for sample sizesn≃14p, maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem.

Keywords

Brownian motion tree model, Convex optimization, Eigenvalues of random matrices, Linear Gaussian covariance model, Tracy–Widom law, Wishart distribution

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