IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i9p2162-d1139369.html
   My bibliography  Save this article

Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix

Author

Listed:
  • Daya K. Nagar

    (Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-108, Medellín 050010, Colombia
    These authors contributed equally to this work.)

  • Alejandro Roldán-Correa

    (Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53-108, Medellín 050010, Colombia
    These authors contributed equally to this work.)

  • Saralees Nadarajah

    (Department of Mathematics, University of Manchester, Manchester M13 9PL, UK
    These authors contributed equally to this work.)

Abstract

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E ( tr ( W h ) ) g ( tr ( W j ) ) i and E ( tr ( W − h ) ) g ( tr ( W − j ) ) i , respectively, for particular values of g , h , i , j , g + h + i + j ≤ 5 , where W follows a complex Wishart distribution. For specific values of g , h , i , j , we first write ( tr ( W h ) ) g ( tr ( W j ) ) i and ( tr ( W − h ) ) g ( tr ( W − j ) ) i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived.

Suggested Citation

  • Daya K. Nagar & Alejandro Roldán-Correa & Saralees Nadarajah, 2023. "Expected Values of Scalar-Valued Functions of a Complex Wishart Matrix," Mathematics, MDPI, vol. 11(9), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2162-:d:1139369
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/9/2162/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/9/2162/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nagar, Daya K. & Roldán-Correa, Alejandro & Gupta, Arjun K., 2013. "Extended matrix variate gamma and beta functions," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 53-69.
    2. Shaman, Paul, 1980. "The inverted complex Wishart distribution and its application to spectral estimation," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 51-59, March.
    3. Hillier, Grant & Kan, Raymond, 2022. "Properties Of The Inverse Of A Noncentral Wishart Matrix," Econometric Theory, Cambridge University Press, vol. 38(6), pages 1092-1116, December.
    4. Letac, Gérard & Massam, Hélène, 2008. "The noncentral Wishart as an exponential family, and its moments," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1393-1417, August.
    5. Dharmawansa, Prathapasinghe & McKay, Matthew R., 2011. "Extreme eigenvalue distributions of some complex correlated non-central Wishart and gamma-Wishart random matrices," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 847-868, April.
    6. Krishnaiah, P. R., 1976. "Some recent developments on complex multivariate distributions," Journal of Multivariate Analysis, Elsevier, vol. 6(1), pages 1-30, March.
    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. Withers, Christopher S. & Nadarajah, Saralees, 2012. "Moments and cumulants for the complex Wishart," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 242-247.
    2. Bhandary, Madhusudan, 1996. "Test for generalized variance in signal processing," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 155-162, April.
    3. Letac, Gérard & Massam, Hélène, 2018. "The Laplace transform (dets)−pexptr(s−1w) and the existence of non-central Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 96-110.
    4. Hélène Massam & Erhard Neher, 1997. "On Transformations and Determinants of Wishart Variables on Symmetric Cones," Journal of Theoretical Probability, Springer, vol. 10(4), pages 867-902, October.
    5. Daya K. Nagar & Raúl Alejandro Morán-Vásquez & Arjun K. Gupta, 2015. "Extended Matrix Variate Hypergeometric Functions and Matrix Variate Distributions," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2015, pages 1-15, January.
    6. 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.
    7. Dharmawansa, Prathapasinghe, 2016. "Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean," Journal of Multivariate Analysis, Elsevier, vol. 149(C), pages 30-53.
    8. Philip L. H. Yu & W. K. Li & F. C. Ng, 2017. "The Generalized Conditional Autoregressive Wishart Model for Multivariate Realized Volatility," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(4), pages 513-527, October.
    9. Chalfant, James & Collender, Robert N. & Subramanfar, Shankar, 1988. "The Mean and Variance of the Mean-Variance Decision Rule," CUDARE Working Papers 198476, University of California, Berkeley, Department of Agricultural and Resource Economics.
    10. Pushpa Narayan Rathie & Luan Carlos de Sena Monteiro Ozelim, 2022. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions," Stats, MDPI, vol. 5(4), pages 1-9, November.
    11. Di Nardo, Elvira, 2014. "On a symbolic representation of non-central Wishart random matrices with applications," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 121-135.
    12. Konno, Yoshihiko, 2007. "Estimation of normal covariance matrices parametrized by irreducible symmetric cones under Stein's loss," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 295-316, February.
    13. Daya K. Nagar & Saralees Nadarajah & Idika E. Okorie, 2017. "A New Bivariate Distribution with One Marginal Defined on the Unit Interval," Annals of Data Science, Springer, vol. 4(3), pages 405-420, September.
    14. Satoshi Kuriki & Yasuhide Numata, 2010. "Graph presentations for moments of noncentral Wishart distributions and their applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(4), pages 645-672, August.
    15. Graczyk, Piotr & Małecki, Jacek & Mayerhofer, Eberhard, 2018. "A characterization of Wishart processes and Wishart distributions," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1386-1404.
    16. Christa Cuchiero & Josef Teichmann, 2011. "Path properties and regularity of affine processes on general state spaces," Papers 1107.1607, arXiv.org, revised Jan 2013.
    17. Chalfant, James A. & Callender, Robert N. & Subramanian, Shankar, 1988. "The Mean And Variance Of The Mean-Variance Decision Rule," Department of Economics and Business - Archive 259434, North Carolina State University, Department of Economics.
    18. 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.
    19. Kołodziejek, Bartosz, 2016. "Characterization of beta distribution on symmetric cones," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 414-423.
    20. Mayerhofer, Eberhard, 2013. "On the existence of non-central Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 448-456.

    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:gam:jmathe:v:11:y:2023:i:9:p:2162-:d:1139369. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.