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Maximum likelihood estimation for linear Gaussian covariance models
Author(s) -
Zwiernik Piotr,
Uhler Caroline,
Richards Donald
Publication year - 2017
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12217
Subject(s) - wishart distribution , mathematics , covariance , estimation of covariance matrices , gaussian , covariance matrix , maximum likelihood sequence estimation , eigenvalues and eigenvectors , mathematical optimization , estimation theory , statistics , multivariate statistics , physics , quantum mechanics
Summary 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 → γ >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 ≃14 p , maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem.