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The Laplace transform (dets)−pexptr(s−1w) and the existence of non-central Wishart distributions

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  • Letac, Gérard
  • Massam, Hélène

Abstract

The problem considered in this paper is to find when the non-central Wishart distribution, defined on the cone Pd¯ of positive semidefinite matrices of order d and with a real-valued shape parameter p, does exist. This can be reduced to the study of the measures m(n,k,d) defined on Pd¯ and with Laplace transform (dets)−n∕2exptr(s−1w), where n is an integer and w=diag(0,…,0,1,…,1) has order d and rank k. Our two main results are the following: we compute m(d−1,d,d) and we show that neither m(d−2,d,d) nor m(d−2,d−1,d) exists. These facts solve the problems of the existence and computation of these non-central Wishart distributions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:163:y:2018:i:c:p:96-110
    DOI: 10.1016/j.jmva.2017.10.005
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    References listed on IDEAS

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    1. 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.
    2. Mayerhofer, Eberhard, 2013. "On the existence of non-central Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 448-456.
    3. M. Casalis & G. Letac, 1994. "Characterization of the Jorgensen set in generalized linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 3(1), pages 145-162, June.
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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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