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Band Width Selection for High Dimensional Covariance Matrix Estimation

Author

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  • Qiu, Yumou
  • Chen, Song Xi

Abstract

The banding estimator of Bickel and Levina (2008a) and its tapering version of Cai, Zhang and Zhou (2010), are important high dimensional covariance estimators. Both estimators require choosing a band width parameter. We propose a band width selector for the banding covariance estimator by minimizing an empirical estimate of the expected squared Frobenius norms of the estimation error matrix. The ratio consistency of the band width selector to the underlying band width is established. We also provide a lower bound for the coverage probability of the underlying band width being contained in an interval around the band width estimate. Extensions to the band width selection for the tapering estimator and threshold level selection for the thresholding covariance estimator are made. Numerical simulations and a case study on sonar spectrum data are conducted to confirm and demonstrate the proposed band width and threshold estimation approaches.

Suggested Citation

  • Qiu, Yumou & Chen, Song Xi, 2014. "Band Width Selection for High Dimensional Covariance Matrix Estimation," MPRA Paper 59641, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:59641
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    References listed on IDEAS

    as
    1. Tony Cai & Weidong Liu & Yin Xia, 2013. "Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 265-277, March.
    2. Jianhua Z. Huang & Naiping Liu & Mohsen Pourahmadi & Linxu Liu, 2006. "Covariance matrix selection and estimation via penalised normal likelihood," Biometrika, Biometrika Trust, vol. 93(1), pages 85-98, March.
    3. Adam J. Rothman & Elizaveta Levina & Ji Zhu, 2010. "A new approach to Cholesky-based covariance regularization in high dimensions," Biometrika, Biometrika Trust, vol. 97(3), pages 539-550.
    4. Wei Biao Wu, 2003. "Nonparametric estimation of large covariance matrices of longitudinal data," Biometrika, Biometrika Trust, vol. 90(4), pages 831-844, December.
    5. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
    6. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, September.
    7. Cai, Tony & Liu, Weidong, 2011. "Adaptive Thresholding for Sparse Covariance Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 672-684.
    8. Bai, Z. D. & Silverstein, Jack W. & Yin, Y. Q., 1988. "A note on the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 166-168, August.
    9. Rothman, Adam J. & Levina, Elizaveta & Zhu, Ji, 2009. "Generalized Thresholding of Large Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 177-186.
    10. Xue, Lingzhou & Zou, Hui, 2013. "Minimax optimal estimation of general bandable covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 45-51.
    11. Yi, Feng & Zou, Hui, 2013. "SURE-tuned tapering estimation of large covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 339-351.
    12. Cai, Tony & Liu, Weidong & Luo, Xi, 2011. "A Constrained â„“1 Minimization Approach to Sparse Precision Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 594-607.
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    Cited by:

    1. Farnè, Matteo & Montanari, Angela, 2020. "A large covariance matrix estimator under intermediate spikiness regimes," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    2. Peng, Liuhua & Chen, Song Xi & Zhou, Wen, 2016. "More powerful tests for sparse high-dimensional covariances matrices," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 124-143.

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    More about this item

    Keywords

    Bandable covariance; Banding estimator; Large $p$; small $n$; Ratio-consistency; Tapering estimator; Thresholding estimator.;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling

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