IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v114y2013icp427-447.html
   My bibliography  Save this article

A subspace estimator for fixed rank perturbations of large random matrices

Author

Listed:
  • Hachem, Walid
  • Loubaton, Philippe
  • Mestre, Xavier
  • Najim, Jamal
  • Vallet, Pascal

Abstract

This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigenspace structure of a certain fixed-rank perturbation of the empirical covariance matrix. An estimation algorithm in the spirit of the well-known MUSIC algorithm for parameter estimation is developed. It relies on an approach recently developed by Benaych-Georges and Nadakuditi (2011) [8,9], relating the eigenspaces of extreme eigenvalues of the empirical covariance matrix with eigenspaces of the perturbation matrix. First and second order analyses of the new algorithm are performed.

Suggested Citation

  • Hachem, Walid & Loubaton, Philippe & Mestre, Xavier & Najim, Jamal & Vallet, Pascal, 2013. "A subspace estimator for fixed rank perturbations of large random matrices," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 427-447.
    • Handle: RePEc:eee:jmvana:v:114:y:2013:i:c:p:427-447
    DOI: 10.1016/j.jmva.2012.08.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X12002047
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2012.08.006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Nadler, Boaz, 2011. "On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 363-371, February.
    2. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    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. Banna, Marwa & Najim, Jamal & Yao, Jianfeng, 2020. "A CLT for linear spectral statistics of large random information-plus-noise matrices," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2250-2281.
    2. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02354596, HAL.
    3. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    4. Couillet, Romain & Pascal, Frédéric & Silverstein, Jack W., 2015. "The random matrix regime of Maronna’s M-estimator with elliptically distributed samples," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 56-78.
    5. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe De Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Documents de travail du Centre d'Economie de la Sorbonne 19022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    6. Emmanuelle Jay & Thibault Soler & Eugénie Terreaux & Jean-Philippe Ovarlez & Frédéric Pascal & Philippe de Peretti & Christophe Chorro, 2019. "Improving portfolios global performance using a cleaned and robust covariance matrix estimate," Post-Print halshs-02354596, HAL.

    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. Deo, Rohit S., 2016. "On the Tracy–Widom approximation of studentized extreme eigenvalues of Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 265-272.
    2. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "PCA consistency for the power spiked model in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 334-354.
    3. Jung, Sungkyu & Sen, Arusharka & Marron, J.S., 2012. "Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 190-203.
    4. Forzani, Liliana & Gieco, Antonella & Tolmasky, Carlos, 2017. "Likelihood ratio test for partial sphericity in high and ultra-high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 18-38.
    5. Michael Bridges & Elizabeth A Heron & Colm O'Dushlaine & Ricardo Segurado & The International Schizophrenia Consortium (ISC) & Derek Morris & Aiden Corvin & Michael Gill & Carlos Pinto, 2011. "Genetic Classification of Populations Using Supervised Learning," PLOS ONE, Public Library of Science, vol. 6(5), pages 1-12, May.
    6. Couillet, Romain, 2015. "Robust spiked random matrices and a robust G-MUSIC estimator," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 139-161.
    7. Joongyeub Yeo & George Papanicolaou, 2016. "Random matrix approach to estimation of high-dimensional factor models," Papers 1611.05571, arXiv.org, revised Nov 2017.
    8. Nick Patterson & Alkes L Price & David Reich, 2006. "Population Structure and Eigenanalysis," PLOS Genetics, Public Library of Science, vol. 2(12), pages 1-20, December.
    9. Benaych-Georges, Florent & Nadakuditi, Raj Rao, 2012. "The singular values and vectors of low rank perturbations of large rectangular random matrices," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 120-135.
    10. Brendan P. W. Ames & Mingyi Hong, 2016. "Alternating direction method of multipliers for penalized zero-variance discriminant analysis," Computational Optimization and Applications, Springer, vol. 64(3), pages 725-754, July.
    11. Patrick K. Kimes & Yufeng Liu & David Neil Hayes & James Stephen Marron, 2017. "Statistical significance for hierarchical clustering," Biometrics, The International Biometric Society, vol. 73(3), pages 811-821, September.
    12. Edoardo Saccenti & Marieke E. Timmerman, 2017. "Considering Horn’s Parallel Analysis from a Random Matrix Theory Point of View," Psychometrika, Springer;The Psychometric Society, vol. 82(1), pages 186-209, March.
    13. Ding, Xiucai & Ji, Hong Chang, 2023. "Spiked multiplicative random matrices and principal components," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 25-60.
    14. Shu Wang & Jia-Ren Lin & Eduardo D Sontag & Peter K Sorger, 2019. "Inferring reaction network structure from single-cell, multiplex data, using toric systems theory," PLOS Computational Biology, Public Library of Science, vol. 15(12), pages 1-25, December.
    15. Feher Kristen & Whelan James & Müller Samuel, 2012. "Exploring Multicollinearity Using a Random Matrix Theory Approach," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(3), pages 1-35, May.
    16. Feldman, Michael J., 2023. "Spiked singular values and vectors under extreme aspect ratios," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    17. Weiming Li & Jianfeng Yao, 2015. "On generalized expectation-based estimation of a population spectral distribution from high-dimensional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 359-373, April.
    18. Bo Zhang & Jiti Gao & Guangming Pan & Yanrong Yang, 2019. "Spiked Eigenvalues of High-Dimensional Separable Sample Covariance Matrices," Monash Econometrics and Business Statistics Working Papers 31/19, Monash University, Department of Econometrics and Business Statistics.
    19. Chiani, Marco, 2014. "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 69-81.
    20. Muni S. Srivastava & Hirokazu Yanagihara & Tatsuya Kubokawa, 2014. "Tests for Covariance Matrices in High Dimension with Less Sample Size," CIRJE F-Series CIRJE-F-933, CIRJE, Faculty of Economics, University of Tokyo.

    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:eee:jmvana:v:114:y:2013:i:c:p:427-447. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.