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The likelihood ratio test for a separable covariance matrix

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  • Lu, Nelson
  • Zimmerman, Dale L.

Abstract

We consider the problem of testing whether a covariance matrix has a separable (Kronecker product) structure. Such structure is of particular interest when the observed variables can be cross-classified by two factors, as occurs for example when comparable or identical characteristics are measured on several parts of each subject. We derive the likelihood ratio test for separability on the basis of a random sample from a multivariate normal population, and we establish an invariance property of the test statistic that allows us to table its null distribution. An example illustrates the methodology.

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  • 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.
    • Handle: RePEc:eee:stapro:v:73:y:2005:i:4:p:449-457
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    1. W. B. Smith & R. R. Hocking, 1972. "Wishart Variate Generator," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 21(3), pages 341-345, November.
    2. Dayanand Naik & Shantha Rao, 2001. "Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix," Journal of Applied Statistics, Taylor & Francis Journals, vol. 28(1), pages 91-105.
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    Cited by:

    1. Timothy Opheim & Anuradha Roy, 2021. "Linear models for multivariate repeated measures data with block exchangeable covariance structure," Computational Statistics, Springer, vol. 36(3), pages 1931-1963, September.
    2. Daniels, M.J. & Pourahmadi, M., 2009. "Modeling covariance matrices via partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2352-2363, November.
    3. Wang, Lili & Paul, Debashis, 2014. "Limiting spectral distribution of renormalized separable sample covariance matrices when p/n→0," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 25-52.
    4. Kohli, Priya & Garcia, Tanya P. & Pourahmadi, Mohsen, 2016. "Modeling the Cholesky factors of covariance matrices of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 87-100.
    5. Filipiak, Katarzyna & Klein, Daniel, 2017. "Estimation of parameters under a generalized growth curve model," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 73-86.
    6. Anuradha Roy & Ricardo Leiva, 2008. "Testing of a Structures Covariance Matrix for Three-Level Repeated Measures Data," Working Papers 0037, College of Business, University of Texas at San Antonio.
    7. Kim, Chulmin & Zimmerman, Dale L., 2012. "Unconstrained models for the covariance structure of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 104-118.
    8. Manceur, A.M. & Dutilleul, P., 2013. "Unbiased modified likelihood ratio tests for simple and double separability of a variance–covariance structure," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 631-636.
    9. Seongoh Park & Johan Lim & Xinlei Wang & Sanghan Lee, 2019. "Permutation based testing on covariance separability," Computational Statistics, Springer, vol. 34(2), pages 865-883, June.
    10. 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.
    11. Guggenberger, Patrik & Kleibergen, Frank & Mavroeidis, Sophocles, 2023. "A test for Kronecker Product Structure covariance matrix," Journal of Econometrics, Elsevier, vol. 233(1), pages 88-112.
    12. Lingzhe Guo & Reza Modarres, 2020. "Testing the equality of matrix distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(2), pages 289-307, June.
    13. Pamela C. Smith & Dana A. Forgione, 2008. "Global Outsourcing of Healthcare: A Medical Tourism Decision Model," Working Papers 0033, College of Business, University of Texas at San Antonio.
    14. Filipiak, Katarzyna & Klein, Daniel & Roy, Anuradha, 2016. "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 105-124.
    15. Kim, Seungkyu & Park, Seongoh & Lim, Johan & Lee, Sang Han, 2023. "Robust tests for scatter separability beyond Gaussianity," Computational Statistics & Data Analysis, Elsevier, vol. 179(C).
    16. Hao, Chengcheng & Liang, Yuli & Mathew, Thomas, 2016. "Testing variance parameters in models with a Kronecker product covariance structure," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 182-189.
    17. 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.
    18. Azaïs, Jean-Marc & Ribes, Aurélien, 2016. "Multivariate spline analysis for multiplicative models: Estimation, testing and application to climate change," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 38-53.
    19. Filipiak, Katarzyna & Klein, Daniel & Mokrzycka, Monika, 2024. "Discrepancy between structured matrices in the power analysis of a separability test," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    20. Katarzyna Filipiak & Daniel Klein & Anuradha Roy, 2015. "Score test for a separable covariance structure with the first component as compound symmetric correlation matrix," Working Papers 0148mss, College of Business, University of Texas at San Antonio.
    21. Liang, Yuli & Hao, Chengcheng & Dai, Deliang, 2024. "Two-sample intraclass correlation coefficient tests for matrix-valued data," Working Papers in Economics and Statistics 6/2024, Linnaeus University, School of Business and Economics, Department of Economics and Statistics.
    22. Martin Ohlson & Zhanna Andrushchenko & Dietrich Rosen, 2011. "Explicit estimators under m-dependence for a multivariate normal distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(1), pages 29-42, February.
    23. Viroli, Cinzia, 2012. "On matrix-variate regression analysis," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 296-309.
    24. Roy, Anuradha & Leiva, Ricardo, 2008. "Likelihood ratio tests for triply multivariate data with structured correlation on spatial repeated measurements," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1971-1980, September.

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