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Abstract
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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
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Mathematical Subject Classification
Primary: 62R01
Secondary: 62F30
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Milestones
Received: 17 January 2024
Revised: 2 June 2024
Accepted: 5 June 2024
Published: 28 August 2024
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© 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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