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Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure

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  • Roś, Beata
  • Bijma, Fetsje
  • de Munck, Jan C.
  • de Gunst, Mathisca C.M.

Abstract

This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. There is no explicit expression for the maximum likelihood estimator of a Kronecker product covariance matrix. We investigate whether the maximum likelihood estimator of the covariance matrix exists and whether it is unique. We consider models with general, with double diagonal, and with one diagonal Kronecker product covariance matrices, and find different results.

Suggested Citation

  • Roś, Beata & Bijma, Fetsje & de Munck, Jan C. & de Gunst, Mathisca C.M., 2016. "Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 345-361.
    • Handle: RePEc:eee:jmvana:v:143:y:2016:i:c:p:345-361
    DOI: 10.1016/j.jmva.2015.05.019
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    References listed on IDEAS

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    1. Lu, Nelson & Zimmerman, Dale L., 2005. "The likelihood ratio test for a separable covariance matrix," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 449-457, July.
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    Cited by:

    1. Soloveychik, I. & Trushin, D., 2016. "Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 92-113.
    2. Gregory Cox, 2018. "Almost Sure Uniqueness of a Global Minimum Without Convexity," Papers 1803.02415, arXiv.org, revised Feb 2019.
    3. Glanz, Hunter & Carvalho, Luis, 2018. "An expectation–maximization algorithm for the matrix normal distribution with an application in remote sensing," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 31-48.
    4. Joni Virta & Niko Lietzén & Pauliina Ilmonen & Klaus Nordhausen, 2021. "Fast tensorial JADE," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 164-187, March.
    5. el Bouhaddani, Said & Uh, Hae-Won & Hayward, Caroline & Jongbloed, Geurt & Houwing-Duistermaat, Jeanine, 2018. "Probabilistic partial least squares model: Identifiability, estimation and application," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 331-346.

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