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October 2021 Existence and uniqueness of the Kronecker covariance MLE
Mathias Drton, Satoshi Kuriki, Peter Hoff
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Ann. Statist. 49(5): 2721-2754 (October 2021). DOI: 10.1214/21-AOS2052


In matrix-valued datasets the sampled matrices often exhibit correlations among both their rows and their columns. A useful and parsimonious model of such dependence is the matrix normal model, in which the covariances among the elements of a random matrix are parameterized in terms of the Kronecker product of two covariance matrices, one representing row covariances and one representing column covariance. An appealing feature of such a matrix normal model is that the Kronecker covariance structure allows for standard likelihood inference even when only a very small number of data matrices is available. For instance, in some cases a likelihood ratio test of dependence may be performed with a sample size of one. However, more generally the sample size required to ensure boundedness of the matrix normal likelihood or the existence of a unique maximizer depends in a complicated way on the matrix dimensions. This motivates the study of how large a sample size is needed to ensure that maximum likelihood estimators exist, and exist uniquely with probability one. Our main result gives precise sample size thresholds in the paradigm where the number of rows and the number of columns of the data matrices differ by at most a factor of two. Our proof uses invariance properties that allow us to consider data matrices in canonical form, as obtained from the Kronecker canonical form for matrix pencils.

Funding Statement

SK was partially supported by JSPS KAKENHI Grant Number JP16H02792.


We are grateful to Satoru Iwata, Lek-Heng Lim and Fumihiro Sato for their comments on the Kronecker canonical form.


Download Citation

Mathias Drton. Satoshi Kuriki. Peter Hoff. "Existence and uniqueness of the Kronecker covariance MLE." Ann. Statist. 49 (5) 2721 - 2754, October 2021.


Received: 1 March 2020; Revised: 1 October 2020; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2052

Primary: 62H12
Secondary: 62R01

Keywords: Gaussian distribution , Kronecker canonical form , matrix normal model , maximum likelihood estimation , separable covariance

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 5 • October 2021
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