Vol. 11, No. 1, 2020

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Estimating linear covariance models with numerical nonlinear algebra

Bernd Sturmfels, Sascha Timme and Piotr Zwiernik

Vol. 11 (2020), No. 1, 31–52
Abstract

Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g., Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.

Keywords
linear covariance model, maximum likelihood, dual maximum likelihood, numerical nonlinear algebra
Mathematical Subject Classification 2010
Primary: 14Q99, 62F10
Milestones
Received: 4 September 2019
Revised: 23 January 2020
Accepted: 30 April 2020
Published: 1 October 2020
Authors
Bernd Sturmfels
Max Planck-Institute for Mathematics in the Sciences
Leipzig
Germany
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Sascha Timme
Institut für Mathematik
Technische Universität Berlin
Berlin
Germany
Piotr Zwiernik
Department of Economics and Business
Universitat Pompeu Fabra
Barcelona
Spain