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On a symbolic representation of non-central Wishart random matrices with applications

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  • Di Nardo, Elvira

Abstract

By using a symbolic method, known in the literature as the classical umbral calculus, the trace of a non-central Wishart random matrix is represented as the convolution of the traces of its central component and of a formal variable matrix. Thanks to this representation, the moments of this random matrix are proved to be a Sheffer polynomial sequence, allowing us to recover several properties. The multivariate symbolic method generalizes the employment of Sheffer representation and a closed form formula for computing joint moments and cumulants (also normalized) is given. By using this closed form formula and a combinatorial device, known in the literature as necklace, an efficient algorithm for their computations is set up. Applications are given to the computation of permanents as well as to the characterization of inherited estimators of cumulants, which turn useful in dealing with minors of non-central Wishart random matrices. An asymptotic approximation of generalized moments involving free probability is proposed.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:125:y:2014:i:c:p:121-135
    DOI: 10.1016/j.jmva.2013.12.001
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    References listed on IDEAS

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    1. Tönu Kollo & Dietrich Rosen, 1995. "Approximating by the Wishart distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 767-783, December.
    2. Kollo, Tõnu & Ruul, Kaire, 2003. "Approximations to the distribution of the sample correlation matrix," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 318-334, May.
    3. 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.
    4. 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.
    5. Withers, Christopher S. & Nadarajah, Saralees, 2012. "Moments and cumulants for the complex Wishart," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 242-247.
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    Cited by:

    1. 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).

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