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Permutation based testing on covariance separability

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Abstract

Separability is an attractive feature of covariance matrices or matrix variate data, which can improve and simplify many multivariate procedures. Due to its importance, testing separability has attracted much attention in the past. The procedures in the literature are of two types, likelihood ratio test (LRT) and Rao’s score test (RST). Both are based on the normality assumption or the large-sample asymptotic properties of the test statistics. In this paper, we develop a new approach that is very different from existing ones. We propose to reformulate the null hypothesis (the separability of a covariance matrix of interest) into many sub-hypotheses (the separability of the sub-matrices of the covariance matrix), which are testable using a permutation based procedure. We then combine the testing results of sub-hypotheses using the Bonferroni and two-stage additive procedures. Our permutation based procedures are inherently distribution free; thus it is robust to non-normality of the data. In addition, unlike the LRT, they are applicable to situations when the sample size is smaller than the number of unknown parameters in the covariance matrix. Our numerical study and data examples show the advantages of our procedures over the existing LRT and RST.

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Acknowledgements

We are grateful to the Associate Editor and two anonymous reviewers for their many helpful comments. The R-package “NPCovSepTest” of the proposed procedure is online available from https://sites.google.com/view/seongohpark/software. J. Lim’s research is supported by National Research Foundation of Korea (Nos. NRF-2017R1A2B2012264; MIST-2011-0030810).

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Authors and Affiliations

  1. Department of Statistics, Seoul National University, Seoul, Korea

    Seongoh Park & Johan Lim

  2. Department of Statistical Science, Southern Methodist University, Dallas, TX, USA

    Xinlei Wang

  3. Center for Biomedical Imaging and Neuromodulation, Nathan Kline Institute for Psychiatric Research, Orangeburg, NY, USA

    Sanghan Lee

Authors
  1. Seongoh Park
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  2. Johan Lim
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  3. Xinlei Wang
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  4. Sanghan Lee
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Corresponding author

Correspondence to Johan Lim.

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Park, S., Lim, J., Wang, X. et al. Permutation based testing on covariance separability. Comput Stat 34, 865–883 (2019). https://doi.org/10.1007/s00180-018-0839-2

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  • DOI: https://doi.org/10.1007/s00180-018-0839-2

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