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Robust estimation of high-dimensional covariance and precision matrices

Author

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  • Marco Avella-Medina
  • Heather S Battey
  • Jianqing Fan
  • Quefeng Li

Abstract

SUMMARYHigh-dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components. Covariance and precision matrices provide a useful summary of such structure, yet the performance of popular matrix estimators typically hinges upon a sub-Gaussianity assumption. This paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions. The proposed estimators, under a bounded fourth moment assumption, achieve the same minimax convergence rates as do existing methods under a sub-Gaussianity assumption. Consistency of the proposed estimators is also established under the weak assumption of bounded $2+\epsilon$ moments for $\epsilon\in (0,2)$. The associated convergence rates depend on $\epsilon$.

Suggested Citation

  • Marco Avella-Medina & Heather S Battey & Jianqing Fan & Quefeng Li, 2018. "Robust estimation of high-dimensional covariance and precision matrices," Biometrika, Biometrika Trust, vol. 105(2), pages 271-284.
    • Handle: RePEc:oup:biomet:v:105:y:2018:i:2:p:271-284.
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    File URL: http://hdl.handle.net/10.1093/biomet/asy011
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    Citations

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    Cited by:

    1. Wang, Xin & Kong, Lingchen & Wang, Liqun, 2024. "Estimation of sparse covariance matrix via non-convex regularization," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    2. Yang, Shuquan & Ling, Nengxiang, 2023. "Robust projected principal component analysis for large-dimensional semiparametric factor modeling," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    3. Kangqiang Li & Han Bao & Lixin Zhang, 2022. "Robust covariance estimation for distributed principal component analysis," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(6), pages 707-732, August.
    4. Xiao, Xuan & Xu, Xingbai & Zhong, Wei, 2023. "Huber estimation for the network autoregressive model," Statistics & Probability Letters, Elsevier, vol. 203(C).
    5. Xin Wang & Lingchen Kong & Liqun Wang & Zhaoqilin Yang, 2023. "High-Dimensional Covariance Estimation via Constrained L q -Type Regularization," Mathematics, MDPI, vol. 11(4), pages 1-20, February.
    6. Li, Kangqiang & Tang, Songqiao & Zhang, Lixin, 2022. "Robust parameter estimation of regression models under weakened moment assumptions," Statistics & Probability Letters, Elsevier, vol. 191(C).
    7. Christis Katsouris, 2021. "Optimal Portfolio Choice and Stock Centrality for Tail Risk Events," Papers 2112.12031, arXiv.org.
    8. Alexander Giessing & Jianqing Fan, 2020. "Bootstrapping $\ell_p$-Statistics in High Dimensions," Papers 2006.13099, arXiv.org, revised Aug 2020.
    9. Zeyu Wu & Cheng Wang & Weidong Liu, 2023. "A unified precision matrix estimation framework via sparse column-wise inverse operator under weak sparsity," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(4), pages 619-648, August.
    10. Li, Huimin & Wang, Jinru, 2024. "Sparse basis covariance matrix estimation for high dimensional compositional data via hard thresholding," Statistics & Probability Letters, Elsevier, vol. 209(C).
    11. Liang, Wanfeng & Wu, Yue & Ma, Xiaoyan, 2022. "Robust sparse precision matrix estimation for high-dimensional compositional data," Statistics & Probability Letters, Elsevier, vol. 184(C).
    12. Zeyu Diao & Lili Yue & Fanrong Zhao & Gaorong Li, 2022. "High-Dimensional Regression Adjustment Estimation for Average Treatment Effect with Highly Correlated Covariates," Mathematics, MDPI, vol. 10(24), pages 1-18, December.

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