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Diagonal nonlinear transformations preserve structure in covariance and precision matrices

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  • Morrison, Rebecca
  • Baptista, Ricardo
  • Basor, Estelle

Abstract

For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision matrices reveal correlations and so-called partial correlations between variables, but these do not, in general, have any correspondence with respect to independence properties. In this paper, we prove that, for a certain class of non-Gaussian distributions, these correspondences still hold, exactly for the covariance and approximately for the precision. The distributions—sometimes referred to as “nonparanormal”—are given by diagonal transformations of multivariate normal random variables. We provide several analytic and numerical examples illustrating these results.

Suggested Citation

  • Morrison, Rebecca & Baptista, Ricardo & Basor, Estelle, 2022. "Diagonal nonlinear transformations preserve structure in covariance and precision matrices," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:jmvana:v:190:y:2022:i:c:s0047259x22000252
    DOI: 10.1016/j.jmva.2022.104983
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    References listed on IDEAS

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    1. Jianqing Fan & Yuan Liao & Han Liu, 2016. "An overview of the estimation of large covariance and precision matrices," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
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