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Moments of a Wishart Matrix

Author

Listed:
  • Grant Hillier

    (CeMMAP, University of Southampton)

  • Raymond Kan

    (University of Toronto)

Abstract

The paper discusses the moments of Wishart matrices, in both the central and noncentral cases. The first part of the paper shows that the expectation map has certain homogeneity and equivariance properties which impose considerable structure on the moments, hitherto unrecognised. The second part of the paper explains how the moments may be computed efficiently. The two parts of the paper are completely independent, but the computations produce precisely the algebraic structure predicted in the first part, as well as reproducing all previously known formulae. A number of examples are given for the more manageable cases.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jqecon:v:19:y:2021:i:1:d:10.1007_s40953-021-00267-7
    DOI: 10.1007/s40953-021-00267-7
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    References listed on IDEAS

    as
    1. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    2. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    3. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2014. "Generating Functions And Short Recursions, With Applications To The Moments Of Quadratic Forms In Noncentral Normal Vectors," Econometric Theory, Cambridge University Press, vol. 30(2), pages 436-473, April.
    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. 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.
    6. Gérard Letac & Hélène 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.
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    Citations

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    Cited by:

    1. Grant Hillier & Raymond M. Kan, 2024. "On the expectations of equivariant matrix‐valued functions of Wishart and inverse Wishart matrices," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 697-723, June.
    2. Yong Bao & Aman Ullah, 2021. "The Special Issue in Honor of Anirudh Lal Nagar: An Introduction," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 1-8, December.
    3. Bailly, Gabriel & von Sachs, Rainer, 2024. "Time-Varying Covariance Matrices Estimation by Nonlinear Wavelet Thresholding in a Log-Euclidean Riemannian Manifold," LIDAM Discussion Papers ISBA 2024004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Yong Bao & Aman Ullah, 2021. "Analytical Finite Sample Econometrics: From A. L. Nagar to Now," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 17-37, December.

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    More about this item

    Keywords

    Wishart matrix; Higher order moments; Homogeneity; Equivariance;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics

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