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Rational maximum likelihood estimators of Kronecker covariance matrices

Mathias Drton, Alexandros Grosdos and Andrew McCormack

Vol. 15 (2024), No. 1, 145–164
Abstract

As is the case for many curved exponential families, the computation of maximum likelihood estimates in a multivariate normal model with a Kronecker covariance structure is typically carried out with an iterative algorithm, specifically, a block-coordinate ascent algorithm. We highlight a setting, specified by a coprime relationship between the sample size and dimension of the Kronecker factors, where the likelihood equations have algebraic degree one and an explicit, easy-to-evaluate rational formula for the maximum likelihood estimator can be found. A partial converse of this result is provided that shows that outside of the aforementioned special setting and for large sample sizes, examples of data sets can be constructed for which the degree of the likelihood equations is larger than one.

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Keywords
Gaussian distribution, Kronecker covariance, matrix normal model, maximum likelihood degree, maximum likelihood estimation
Mathematical Subject Classification
Primary: 62R01
Secondary: 62F30
Milestones
Received: 17 January 2024
Revised: 2 June 2024
Accepted: 5 June 2024
Published: 28 August 2024
Authors
Mathias Drton
Department of Mathematics
Technical University of Munich
Munich
Germany
Alexandros Grosdos
Faculty of Mathematics
University of Augsburg
Augsburg
Germany
Andrew McCormack
Department of Mathematics
Technical University of Munich
Munich
Germany