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A useful variant of the Davis–Kahan theorem for statisticians

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  • Y. Yu
  • T. Wang
  • R. J. Samworth

Abstract

The Davis–Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. It relies on an eigenvalue separation condition between certain population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, making it more natural and convenient for direct application in statistical contexts, and provide an improvement in many cases to the usual bound in the statistical literature. We also give an extension to situations where the matrices under study may be asymmetric or even non-square, and where interest is in the distance between subspaces spanned by corresponding singular vectors.

Suggested Citation

  • Y. Yu & T. Wang & R. J. Samworth, 2015. "A useful variant of the Davis–Kahan theorem for statisticians," Biometrika, Biometrika Trust, vol. 102(2), pages 315-323.
    • Handle: RePEc:oup:biomet:v:102:y:2015:i:2:p:315-323.
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    File URL: http://hdl.handle.net/10.1093/biomet/asv008
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    Cited by:

    1. Chao, Shih-Kang & Härdle, Wolfgang K. & Yuan, Ming, 2021. "Factorisable Multitask Quantile Regression," Econometric Theory, Cambridge University Press, vol. 37(4), pages 794-816, August.
    2. Matteo Barigozzi, 2023. "Asymptotic equivalence of Principal Components and Quasi Maximum Likelihood estimators in Large Approximate Factor Models," Papers 2307.09864, arXiv.org, revised Sep 2023.
    3. Barigozzi, Matteo & Cho, Haeran & Fryzlewicz, Piotr, 2018. "Simultaneous multiple change-point and factor analysis for high-dimensional time series," Journal of Econometrics, Elsevier, vol. 206(1), pages 187-225.
    4. Yu, Long & He, Yong & Kong, Xinbing & Zhang, Xinsheng, 2022. "Projected estimation for large-dimensional matrix factor models," Journal of Econometrics, Elsevier, vol. 229(1), pages 201-217.
    5. Long Zhao & Deepayan Chakrabarti & Kumar Muthuraman, 2019. "Portfolio Construction by Mitigating Error Amplification: The Bounded-Noise Portfolio," Operations Research, INFORMS, vol. 67(4), pages 965-983, July.
    6. Wang, Wuyi & Su, Liangjun, 2021. "Identifying latent group structures in nonlinear panels," Journal of Econometrics, Elsevier, vol. 220(2), pages 272-295.
    7. Matteo Barigozzi, 2022. "On Estimation and Inference of Large Approximate Dynamic Factor Models via the Principal Component Analysis," Papers 2211.01921, arXiv.org, revised Jul 2023.
    8. Denis Chetverikov & Elena Manresa, 2022. "Spectral and post-spectral estimators for grouped panel data models," Papers 2212.13324, arXiv.org, revised Dec 2022.
    9. Milbradt, Cassandra & Wahl, Martin, 2020. "High-probability bounds for the reconstruction error of PCA," Statistics & Probability Letters, Elsevier, vol. 161(C).
    10. Fogel, Fajwel & d'Aspremont, Alexandre & Vojnovic, Milan, 2016. "Spectral ranking using seriation," LSE Research Online Documents on Economics 68987, London School of Economics and Political Science, LSE Library.
    11. Steland, Ansgar, 2020. "Testing and estimating change-points in the covariance matrix of a high-dimensional time series," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    12. Timothy I. Cannings & Richard J. Samworth, 2017. "Random-projection ensemble classification," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(4), pages 959-1035, September.
    13. Banerjee, Sayantan & Akbani, Rehan & Baladandayuthapani, Veerabhadran, 2019. "Spectral clustering via sparse graph structure learning with application to proteomic signaling networks in cancer," Computational Statistics & Data Analysis, Elsevier, vol. 132(C), pages 46-69.
    14. Avanti Athreya & Joshua Cape & Minh Tang, 2022. "Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 36-63, June.

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