IDEAS home Printed from https://ideas.repec.org/p/tky/fseres/2015cf995.html
   My bibliography  Save this paper

Linear Ridge Estimator of High-Dimensional Precision Matrix Using Random Matrix Theory

Author

Listed:
  • Tsubasa Ito

    (Graduate School of Economics, The University of Tokyo)

  • Tatsuya Kubokawa

    (Faculty of Economics, The University of Tokyo)

Abstract

In estimation of the large precision matrix, this paper suggests a new shrinkage estimator, called the linear ridge estimator. This estimator is motivated from a Bayesian aspect for a spike and slab prior distribution of the precision matrix, and has a form of convex combination of the ridge estimator and the identity matrix multiplied by scalar. The optimal parameters in the linear ridge estimator are derived in terms of minimizing a Frobenius loss function and estimated in closed forms based on the random matrix theory. Finally, the performance of the linear ridge estimator is numerically investigated and compared with some existing estimators.

Suggested Citation

  • Tsubasa Ito & Tatsuya Kubokawa, 2015. "Linear Ridge Estimator of High-Dimensional Precision Matrix Using Random Matrix Theory ," CIRJE F-Series CIRJE-F-995, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2015cf995
    as

    Download full text from publisher

    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2015/2015cf995.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Ledoit, Olivier & Wolf, Michael, 2015. "Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 360-384.
    3. Jushan Bai & Shuzhong Shi, 2011. "Estimating High Dimensional Covariance Matrices and its Applications," Annals of Economics and Finance, Society for AEF, vol. 12(2), pages 199-215, November.
    4. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2016. "Direct shrinkage estimation of large dimensional precision matrix," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 223-236.
    5. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    6. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    7. Touloumis, Anestis, 2015. "Nonparametric Stein-type shrinkage covariance matrix estimators in high-dimensional settings," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 251-261.
    8. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
    9. Fisher, Thomas J. & Sun, Xiaoqian, 2011. "Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1909-1918, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ledoit, Olivier & Wolf, Michael, 2017. "Numerical implementation of the QuEST function," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 199-223.
    2. Jonathan Gillard & Emily O’Riordan & Anatoly Zhigljavsky, 2023. "Polynomial whitening for high-dimensional data," Computational Statistics, Springer, vol. 38(3), pages 1427-1461, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Olivier Ledoit & Michael Wolf, 2019. "Quadratic shrinkage for large covariance matrices," ECON - Working Papers 335, Department of Economics - University of Zurich, revised Dec 2020.
    2. Ledoit, Olivier & Wolf, Michael, 2017. "Numerical implementation of the QuEST function," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 199-223.
    3. Bodnar, Olha & Bodnar, Taras & Parolya, Nestor, 2022. "Recent advances in shrinkage-based high-dimensional inference," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    4. Robert F. Engle & Olivier Ledoit & Michael Wolf, 2019. "Large Dynamic Covariance Matrices," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(2), pages 363-375, April.
    5. Olivier Ledoit & Michael Wolf, 2013. "Optimal estimation of a large-dimensional covariance matrix under Stein’s loss," ECON - Working Papers 122, Department of Economics - University of Zurich, revised Mar 2017.
    6. Wen, Jun, 2018. "Estimation of two high-dimensional covariance matrices and the spectrum of their ratio," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 1-29.
    7. Ikeda, Yuki & Kubokawa, Tatsuya & Srivastava, Muni S., 2016. "Comparison of linear shrinkage estimators of a large covariance matrix in normal and non-normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 95(C), pages 95-108.
    8. Olivier Ledoit & Michael Wolf, 2017. "Analytical nonlinear shrinkage of large-dimensional covariance matrices," ECON - Working Papers 264, Department of Economics - University of Zurich, revised Nov 2018.
    9. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2018. "Estimation of the global minimum variance portfolio in high dimensions," European Journal of Operational Research, Elsevier, vol. 266(1), pages 371-390.
    10. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    11. Bodnar, Taras & Gupta, Arjun K. & Parolya, Nestor, 2014. "On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 215-228.
    12. Taras Bodnar & Arjun K. Gupta & Nestor Parolya, 2013. "Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix," Papers 1308.0931, arXiv.org, revised Mar 2014.
    13. Yuasa, Ryota & Kubokawa, Tatsuya, 2020. "Ridge-type linear shrinkage estimation of the mean matrix of a high-dimensional normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 178(C).
    14. Yang, Guangren & Liu, Yiming & Pan, Guangming, 2019. "Weighted covariance matrix estimation," Computational Statistics & Data Analysis, Elsevier, vol. 139(C), pages 82-98.
    15. Couillet, Romain & Kammoun, Abla & Pascal, Frédéric, 2016. "Second order statistics of robust estimators of scatter. Application to GLRT detection for elliptical signals," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 249-274.
    16. Jamshid Namdari & Debashis Paul & Lili Wang, 2021. "High-Dimensional Linear Models: A Random Matrix Perspective," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 645-695, August.
    17. Couillet, Romain & McKay, Matthew, 2014. "Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 99-120.
    18. Olivier Ledoit & Sandrine P�ch�, 2009. "Eigenvectors of some large sample covariance matrices ensembles," IEW - Working Papers 407, Institute for Empirical Research in Economics - University of Zurich.
    19. Ruili Sun & Tiefeng Ma & Shuangzhe Liu & Milind Sathye, 2019. "Improved Covariance Matrix Estimation for Portfolio Risk Measurement: A Review," JRFM, MDPI, vol. 12(1), pages 1-34, March.
    20. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:tky:fseres:2015cf995. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CIRJE administrative office (email available below). General contact details of provider: https://edirc.repec.org/data/ritokjp.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.