IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2304.07045.html

Ledoit-Wolf linear shrinkage with unknown mean

Author

Listed:
  • Benoit Oriol
  • Alexandre Miot

Abstract

This work addresses large dimensional covariance matrix estimation with unknown mean. The empirical covariance estimator fails when dimension and number of samples are proportional and tend to infinity, settings known as Kolmogorov asymptotics. When the mean is known, Ledoit and Wolf (2004) proposed a linear shrinkage estimator and proved its convergence under those asymptotics. To the best of our knowledge, no formal proof has been proposed when the mean is unknown. To address this issue, we propose to extend the linear shrinkage and its convergence properties to translation-invariant estimators. We expose four estimators respecting those conditions, proving their properties. Finally, we show empirically that a new estimator we propose outperforms other standard estimators.

Suggested Citation

  • Benoit Oriol & Alexandre Miot, 2023. "Ledoit-Wolf linear shrinkage with unknown mean," Papers 2304.07045, arXiv.org, revised Mar 2025.
    • Handle: RePEc:arx:papers:2304.07045
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2304.07045
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Olivier Ledoit & Michael Wolf, 2019. "The power of (non-)linear shrinking: a review and guide to covariance matrix estimation," ECON - Working Papers 323, Department of Economics - University of Zurich, revised Feb 2020.
    3. 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.
    4. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," The Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    5. Ikeda, Yuki & Kubokawa, Tatsuya, 2016. "Linear shrinkage estimation of large covariance matrices using factor models," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 61-81.
    6. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    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. Oriol, Benoît & Miot, Alexandre, 2025. "Ledoit-Wolf linear shrinkage with unknown mean," Journal of Multivariate Analysis, Elsevier, vol. 208(C).
    2. Ortiz, Roberto & Contreras, Mauricio & Mellado, Cristhian, 2023. "Regression, multicollinearity and Markowitz," Finance Research Letters, Elsevier, vol. 58(PC).
    3. Thomas Gehrig & Leopold Sögner & Arne Westerkamp, 2025. "Extending the demand system approach to asset pricing," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 39(1), pages 133-166, March.
    4. Weilong Liu & Yanchu Liu, 2025. "Covariance Matrix Estimation for Positively Correlated Assets," Papers 2507.01545, arXiv.org.
    5. Lauren Stagnol, 2015. "Designing a corporate bond index on solvency criteria," EconomiX Working Papers 2015-39, University of Paris Nanterre, EconomiX.
    6. Liusha Yang & Romain Couillet & Matthew R. McKay, 2015. "A Robust Statistics Approach to Minimum Variance Portfolio Optimization," Papers 1503.08013, arXiv.org.
    7. McDowell, Shaun, 2018. "An empirical evaluation of estimation error reduction strategies applied to international diversification," Journal of Multinational Financial Management, Elsevier, vol. 44(C), pages 1-13.
    8. Füss, Roland & Miebs, Felix & Trübenbach, Fabian, 2014. "A jackknife-type estimator for portfolio revision," Journal of Banking & Finance, Elsevier, vol. 43(C), pages 14-28.
    9. Tae-Hwy Lee & Ekaterina Seregina, 2024. "Optimal Portfolio Using Factor Graphical Lasso," Journal of Financial Econometrics, Oxford University Press, vol. 22(3), pages 670-695.
    10. Paolella, Marc S. & Polak, Paweł & Walker, Patrick S., 2021. "A non-elliptical orthogonal GARCH model for portfolio selection under transaction costs," Journal of Banking & Finance, Elsevier, vol. 125(C).
    11. Yi Huang & Wei Zhu & Duan Li & Shushang Zhu & Shikun Wang, 2023. "Integrating Different Informations for Portfolio Selection," Papers 2305.17881, arXiv.org.
    12. Jun Yuan & Qi Xu & Ying Wang, 2023. "Probability weighting in commodity futures markets," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(4), pages 516-548, April.
    13. Istvan Varga-Haszonits & Fabio Caccioli & Imre Kondor, 2016. "Replica approach to mean-variance portfolio optimization," Papers 1606.08679, arXiv.org.
    14. Santos, André A.P. & Torrent, Hudson S., 2022. "Markowitz meets technical analysis: Building optimal portfolios by exploiting information in trend-following signals," Finance Research Letters, Elsevier, vol. 49(C).
    15. Guillaume Coqueret, 2016. "Empirical properties of a heterogeneous agent model in large dimensions," Post-Print hal-02088097, HAL.
    16. Thomas Trier Bjerring & Omri Ross & Alex Weissensteiner, 2017. "Feature selection for portfolio optimization," Annals of Operations Research, Springer, vol. 256(1), pages 21-40, September.
    17. Bollerslev, Tim & Patton, Andrew J. & Quaedvlieg, Rogier, 2018. "Modeling and forecasting (un)reliable realized covariances for more reliable financial decisions," Journal of Econometrics, Elsevier, vol. 207(1), pages 71-91.
    18. Vincent Tan & Stefan Zohren, 2020. "Estimation of Large Financial Covariances: A Cross-Validation Approach," Papers 2012.05757, arXiv.org, revised Jan 2023.
    19. 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.
    20. Vrinda Dhingra & S. K. Gupta, 2025. "An Empirical Study of Robust Mean-Variance Portfolios with Short Selling," Computational Economics, Springer;Society for Computational Economics, vol. 66(3), pages 1943-1968, September.

    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:arx:papers:2304.07045. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.