IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2303.18233.html
   My bibliography  Save this paper

Inference on eigenvectors of non-symmetric matrices

Author

Listed:
  • Jerome R. Simons

Abstract

This paper argues that the symmetrisability condition in Tyler (1981) is not necessary to establish asymptotic inference procedures for eigenvectors. We establish distribution theory for a Wald and t-test for full-vector and individual coefficient hypotheses, respectively. Our test statistics originate from eigenprojections of non-symmetric matrices. Representing projections as a mapping from the underlying matrix to its spectral data, we find derivatives through analytic perturbation theory. These results demonstrate how the analytic perturbation theory of Sun (1991) is a useful tool in multivariate statistics and are of independent interest. As an application, we define confidence sets for Bonacich centralities estimated from adjacency matrices induced by directed graphs.

Suggested Citation

  • Jerome R. Simons, 2023. "Inference on eigenvectors of non-symmetric matrices," Papers 2303.18233, arXiv.org, revised Apr 2023.
    • Handle: RePEc:arx:papers:2303.18233
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2303.18233
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Takemura, Akimichi & Sheena, Yo, 2005. "Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 271-299, June.
    2. Bura, E. & Pfeiffer, R., 2008. "On the distribution of the left singular vectors of a random matrix and its applications," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2275-2280, October.
    3. Anderson, T.W., 2010. "The LIML estimator has finite moments!," Journal of Econometrics, Elsevier, vol. 157(2), pages 359-361, August.
    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. Eric Blankmeyer, 2020. "NISE Estimation of an Economic Model of Crime," Papers 2003.07860, arXiv.org.
    2. Peter D. Hoff, 2009. "A hierarchical eigenmodel for pooled covariance estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 971-992, November.
    3. Carrasco, Marine & Tchuente, Guy, 2015. "Regularized LIML for many instruments," Journal of Econometrics, Elsevier, vol. 186(2), pages 427-442.
    4. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2013. "Nonparametric estimation of finite mixtures," SciencePo Working papers hal-00972868, HAL.
    5. Koki Shimizu & Hiroki Hashiguchi, 2024. "Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix," Mathematics, MDPI, vol. 12(6), pages 1-11, March.
    6. Hashiguchi, Hiroki & Numata, Yasuhide & Takayama, Nobuki & Takemura, Akimichi, 2013. "The holonomic gradient method for the distribution function of the largest root of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 296-312.
    7. Sheena, Yo & Takemura, Akimichi, 2008. "Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 751-775, April.
    8. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Nonparametric estimation of finite measures," CeMMAP working papers 11/14, Institute for Fiscal Studies.
    9. Seung Jun Shin & Yichao Wu & Hao Helen Zhang & Yufeng Liu, 2017. "Principal weighted support vector machines for sufficient dimension reduction in binary classification," Biometrika, Biometrika Trust, vol. 104(1), pages 67-81.
    10. Stéphane Bonhomme & Koen Jochmans & Jean-Marc Robin, 2014. "Nonparametric spectral-based estimation of latent structures," CeMMAP working papers CWP18/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. repec:spo:wpmain:info:hdl:2441/7o52iohb7k6srk09n8t4k21sm is not listed on IDEAS
    12. repec:hal:wpspec:info:hdl:2441/7o52iohb7k6srk09n8t4k21sm is not listed on IDEAS
    13. Gareth Liu-Evans & Garry DA Phillips, 2023. "The Bias of the Modified Limited Information Maximum Likelihood Estimator (MLIML) in Static Simultaneous Equation Models," Working Papers 202303, University of Liverpool, Department of Economics.
    14. Bura, E. & Yang, J., 2011. "Dimension estimation in sufficient dimension reduction: A unifying approach," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 130-142, January.
    15. Jang, Hyun Jung & Shin, Seung Jun & Artemiou, Andreas, 2023. "Principal weighted least square support vector machine: An online dimension-reduction tool for binary classification," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    16. repec:spo:wpecon:info:hdl:2441/7o52iohb7k6srk09n8t4k21sm is not listed on IDEAS
    17. Sheena, Yo & Takemura, Akimichi, 2011. "Admissible estimator of the eigenvalues of the variance-covariance matrix for multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 801-815, April.
    18. repec:hal:spmain:info:hdl:2441/7o52iohb7k6srk09n8t4k21sm is not listed on IDEAS
    19. Blankmeyer, Eric, 2021. "Explorations in NISE Estimation," MPRA Paper 108179, University Library of Munich, Germany.
    20. Phillips, Garry D.A. & Liu-Evans, Gareth, 2016. "Approximating and reducing bias in 2SLS estimation of dynamic simultaneous equation models," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 734-762.

    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:2303.18233. 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.