IDEAS home Printed from https://ideas.repec.org/a/bla/scjsta/v31y2004i2p295-318.html
   My bibliography  Save this article

All Invariant Moments of the Wishart Distribution

Author

Listed:
  • GĂ©rard Letac
  • HĂ©lĂšne Massam

Abstract

. In this paper, we compute moments of a Wishart matrix variate U of the form đ”Œ(Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U− 1. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U. This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.

Suggested Citation

  • GĂ©rard Letac & HĂ©lĂšne Massam, 2004. "All Invariant Moments of the Wishart Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(2), pages 295-318, June.
    • Handle: RePEc:bla:scjsta:v:31:y:2004:i:2:p:295-318
    DOI: 10.1111/j.1467-9469.2004.01-043.x
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/j.1467-9469.2004.01-043.x
    Download Restriction: no
    File URL: https://libkey.io/10.1111/j.1467-9469.2004.01-043.x?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
    ---><---

    Citations

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


    Cited by:

    1. Sho Matsumoto, 2012. "General Moments of the Inverse Real Wishart Distribution and Orthogonal Weingarten Functions," Journal of Theoretical Probability, Springer, vol. 25(3), pages 798-822, September.
    2. Anatolyev, Stanislav, 2012. "Inference in regression models with many regressors," Journal of Econometrics, Elsevier, vol. 170(2), pages 368-382.
    3. van Wieringen, Wessel N., 2017. "On the mean squared error of the ridge estimator of the covariance and precision matrix," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 88-92.
    4. Zongliang Hu & Zhishui Hu & Kai Dong & Tiejun Tong & Yuedong Wang, 2021. "A shrinkage approach to joint estimation of multiple covariance matrices," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 339-374, April.
    5. van Wieringen, Wessel N. & Peeters, Carel F.W., 2016. "Ridge estimation of inverse covariance matrices from high-dimensional data," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 284-303.
    6. K. Triantafyllopoulos, 2008. "Multivariate stochastic volatility using state space models," Papers 0802.0223, arXiv.org.
    7. Ouimet, Frédéric, 2022. "A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    8. Kozubowski, Tomasz J. & Mazur, Stepan & Podgórski, Krzysztof, 2022. "Matrix Gamma Distributions and Related Stochastic Processes," Working Papers 2022:12, Örebro University, School of Business.
    9. Marcos Escobar & Sven Panz, 2016. "A Note on the Impact of Parameter Uncertainty on Barrier Derivatives," Risks, MDPI, vol. 4(4), pages 1-25, September.
    10. Kubokawa, Tatsuya & Hyodo, Masashi & Srivastava, Muni S., 2013. "Asymptotic expansion and estimation of EPMC for linear classification rules in high dimension," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 496-515.
    11. Nardo, Elvira Di, 2020. "Polynomial traces and elementary symmetric functions in the latent roots of a non-central Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    12. Letac, GĂ©rard & Massam, HĂ©lĂšne, 2008. "The noncentral Wishart as an exponential family, and its moments," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1393-1417, August.
    13. Grant Hillier & Raymond Kan, 2021. "Moments of a Wishart Matrix," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 141-162, December.
    14. Jin-Ting Zhang & Xuefeng Liu, 2013. "A modified Bartlett test for heteroscedastic one-way MANOVA," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 135-152, January.
    15. Klein, Daniel & Pielaszkiewicz, Jolanta & Filipiak, Katarzyna, 2022. "Approximate normality in testing an exchangeable covariance structure under large- and high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    16. Tatsuya Kubokawa & Masashi Hyodo & Muni S. Srivastava, 2011. "Asymptotic Expansion and Estimation of EPMC for Linear Classification Rules in High Dimension," CIRJE F-Series CIRJE-F-818, CIRJE, Faculty of Economics, University of Tokyo.

    More about this item

    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:bla:scjsta:v:31:y:2004:i:2:p:295-318. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0303-6898 .

    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.