Maximum likelihood estimation for linear Gaussian covariance models

article Maximum likelihood estimation for linear Gaussian covariance models published yes Piotr Zwiernik author Caroline Uhler author 49ADD78E-F248-11E8-B48F-1D18A9856A870000-0002-7008-0216 Donald Richards author CaUh department Gaussian Graphical Models: Theory and Applications project 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≃14p, maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem. © 2016 The Royal Statistical Society and Blackwell Publishing Ltd. Wiley-Blackwell2017 eng Journal of the Royal Statistical Society. Series B: Statistical Methodology 1369-7412 1408.5604 00041171230001210.1111/rssb.12217 7941269 - 1292 Zwiernik, P., Uhler, C., & Richards, D. (2017). Maximum likelihood estimation for linear Gaussian covariance models. Journal of the Royal Statistical Society. Series B: Statistical Methodology. Wiley-Blackwell. <a href="https://doi.org/10.1111/rssb.12217">https://doi.org/10.1111/rssb.12217</a> Zwiernik P, Uhler C, Richards D. 2017. Maximum likelihood estimation for linear Gaussian covariance models. Journal of the Royal Statistical Society. Series B: Statistical Methodology. 79(4), 1269–1292. P. Zwiernik, C. Uhler, D. Richards, Journal of the Royal Statistical Society. Series B: Statistical Methodology 79 (2017) 1269–1292. Zwiernik, Piotr, Caroline Uhler, and Donald Richards. “Maximum Likelihood Estimation for Linear Gaussian Covariance Models.” Journal of the Royal Statistical Society. Series B: Statistical Methodology. Wiley-Blackwell, 2017. <a href="https://doi.org/10.1111/rssb.12217">https://doi.org/10.1111/rssb.12217</a>. P. Zwiernik, C. Uhler, and D. Richards, “Maximum likelihood estimation for linear Gaussian covariance models,” Journal of the Royal Statistical Society. Series B: Statistical Methodology, vol. 79, no. 4. Wiley-Blackwell, pp. 1269–1292, 2017. Zwiernik, Piotr, et al. “Maximum Likelihood Estimation for Linear Gaussian Covariance Models.” Journal of the Royal Statistical Society. Series B: Statistical Methodology, vol. 79, no. 4, Wiley-Blackwell, 2017, pp. 1269–92, doi:<a href="https://doi.org/10.1111/rssb.12217">10.1111/rssb.12217</a>. Zwiernik P, Uhler C, Richards D. Maximum likelihood estimation for linear Gaussian covariance models. Journal of the Royal Statistical Society Series B: Statistical Methodology. 2017;79(4):1269-1292. doi:<a href="https://doi.org/10.1111/rssb.12217">10.1111/rssb.12217</a> 12082018-12-11T11:50:43Z2025-06-04T09:41:22Z