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Exact test theory in Gaussian graphical models

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  • Bodnar, Olha
  • Touli, Elena Farahbakhsh

Abstract

In this paper, we derive several statistical tests on the precision matrix with application to the determination of the structure of an undirected Gaussian graph. The exact distributions of the test statistics are obtained under the null hypotheses, while the exact distributions of the random matrices, which are used in the construction of the test statistics, are deduced under the alternative hypothesis. Moreover, we present the high-dimensional asymptotic distributions of the test statistics under the null hypothesis. The testing problems that an undirected Gaussian graph possesses a structure that corresponds to the precision matrix of an AR(1) process, to the block-diagonal precision matrix and to the precision of a factor model are discussed in detail. The performance of the proposed statistical tests is further investigated via an extensive simulation study and compared to the benchmark approach.

Suggested Citation

  • Bodnar, Olha & Touli, Elena Farahbakhsh, 2023. "Exact test theory in Gaussian graphical models," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:jmvana:v:196:y:2023:i:c:s0047259x23000313
    DOI: 10.1016/j.jmva.2023.105185
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    References listed on IDEAS

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    1. Bodnar, Taras & Reiß, Markus, 2016. "Exact and asymptotic tests on a factor model in low and large dimensions with applications," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 125-151.
    2. Bodnar, Taras & Okhrin, Yarema, 2008. "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2389-2405, November.
    3. Bodnar, Taras & Dette, Holger & Parolya, Nestor, 2019. "Testing for independence of large dimensional vectors," MPRA Paper 97997, University Library of Munich, Germany, revised May 2019.
    4. Yang Ni & Veerabhadran Baladandayuthapani & Marina Vannucci & Francesco C. Stingo, 2022. "Bayesian graphical models for modern biological applications," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(2), pages 197-225, June.
    5. Kakizawa, Yoshihide & Iwashita, Toshiya, 2008. "A comparison of higher-order local powers of a class of one-way MANOVA tests under general distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1128-1153, July.
    6. Yang Ni & Veerabhadran Baladandayuthapani & Marina Vannucci & Francesco C. Stingo, 2022. "Rejoinder to the discussion of “Bayesian graphical models for modern biological applications”," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(2), pages 287-294, June.
    7. Schott, James R., 2007. "Some high-dimensional tests for a one-way MANOVA," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1825-1839, October.
    8. Mathias Drton, 2004. "Model selection for Gaussian concentration graphs," Biometrika, Biometrika Trust, vol. 91(3), pages 591-602, September.
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