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Equality tests of covariance matrices under a low-dimensional factor structure

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  • Hyodo, Masashi
  • Nishiyama, Takahiro
  • Watanabe, Hiroki
  • Nakagawa, Tomoyuki
  • Tahata, Kouji

Abstract

We propose an equality test to compare two covariance matrices in a high-dimensional framework while accommodating a low-dimensional latent factor model. We show that null limiting distributions of the test statistics follow a weighted mixture of chi-square distributions under a high-dimensional asymptotic regime combined with weak technical conditions. This distribution depends on the noise covariance matrix and the number of latent factors. Because latent factors are often unknown, we employ an estimation that builds on recent advances in random matrix theory. A numerical study demonstrates the asymptotic power of the proposed test and confirms its favorable analytical properties compared to existing procedures.

Suggested Citation

  • Hyodo, Masashi & Nishiyama, Takahiro & Watanabe, Hiroki & Nakagawa, Tomoyuki & Tahata, Kouji, 2025. "Equality tests of covariance matrices under a low-dimensional factor structure," Journal of Multivariate Analysis, Elsevier, vol. 206(C).
    • Handle: RePEc:eee:jmvana:v:206:y:2025:i:c:s0047259x24001040
    DOI: 10.1016/j.jmva.2024.105397
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    References listed on IDEAS

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    1. H. Wang, 2012. "Factor profiled sure independence screening," Biometrika, Biometrika Trust, vol. 99(1), pages 15-28.
    2. Hyodo, Masashi & Nishiyama, Takahiro & Pavlenko, Tatjana, 2023. "A Behrens–Fisher problem for general factor models in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    3. Ma, Yingying & Lan, Wei & Wang, Hansheng, 2015. "A high dimensional two-sample test under a low dimensional factor structure," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 162-170.
    4. Seung C. Ahn & Alex R. Horenstein, 2013. "Eigenvalue Ratio Test for the Number of Factors," Econometrica, Econometric Society, vol. 81(3), pages 1203-1227, May.
    5. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    6. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    7. Chung Dongjun & Keles Sunduz, 2010. "Sparse Partial Least Squares Classification for High Dimensional Data," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 9(1), pages 1-32, March.
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