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Maximum Likelihood Estimation for Linear Gaussian Covariance Models
We study parameter estimation in linear Gaussian covariance models, which are
-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 , the proofs of our results utilize large-sample
asymptotic theory under the scheme . Remarkably, our
numerical simulations indicate that our results remain valid for as small
as . An important consequence of this analysis is that for sample sizes , maximum likelihood estimation for linear Gaussian covariance
models behaves as if it were a convex optimization problem
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