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The noncentral Wishart as an exponential family, and its moments

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  • Letac, Gérard
  • Massam, Hélène

Abstract

While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix where Y1,...,Yn are independent Gaussian rows in with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by [gamma](p,a;[sigma]) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix [sigma] of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix [omega]=[sigma]a[sigma] is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of when X~[gamma](p,a,[sigma]) and h1,...,hm are arbitrary symmetric matrices of order k, the estimation of the parameters (a,[sigma]) by a method different from that of Alam and Mitra [K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhya, Series B 52 (1990) 133-143] and the determination of the set of acceptable p's as already done by Gindikin and Shanbag for the ordinary Wishart distribution [gamma](p,0,[sigma]).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:7:p:1393-1417
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    References listed on IDEAS

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    1. Leung, Pui Lam, 1994. "An identity for the noncentral wishart distribution with application," Journal of Multivariate Analysis, Elsevier, vol. 48(1), pages 107-114, January.
    2. 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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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Christa Cuchiero & Josef Teichmann, 2011. "Path properties and regularity of affine processes on general state spaces," Papers 1107.1607, arXiv.org, revised Jan 2013.
    5. Mayerhofer, Eberhard, 2013. "On the existence of non-central Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 448-456.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.

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