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The Matsumoto-Yor property and the structure of the Wishart distribution

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  • Massam, Hélène
  • Wesolowski, Jacek

Abstract

This paper establishes a link between a generalized matrix Matsumoto-Yor (MY) property and the Wishart distribution. This link highlights certain conditional independence properties within blocks of the Wishart and leads to a new characterization of the Wishart distribution similar to the one recently obtained by Geiger and Heckerman but involving independences for only three pairs of block partitionings of the random matrix. In the process, we obtain two other main results. The first one is an extension of the MY independence property to random matrices of different dimensions. The second result is its converse. It extends previous characterizations of the matrix generalized inverse Gaussian and Wishart seen as a couple of distributions. We present two proofs for the generalized MY property. The first proof relies on a new version of Herz's identity for Bessel functions of matrix arguments. The second proof uses a representation of the MY property through the structure of the Wishart.

Suggested Citation

  • Massam, Hélène & Wesolowski, Jacek, 2006. "The Matsumoto-Yor property and the structure of the Wishart distribution," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 103-123, January.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:1:p:103-123
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    References listed on IDEAS

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    1. Ronald W. Butler, 1998. "Generalized Inverse Gaussian Distributions and their Wishart Connections," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 69-75, March.
    2. Matsumoto, Hiroyuki & Yor, Marc, 2003. "Interpretation via Brownian motion of some independence properties between GIG and gamma variables," Statistics & Probability Letters, Elsevier, vol. 61(3), pages 253-259, February.
    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. Matsumoto, Hiroyuki & Wesolowski, Jacek & Witkowski, Piotr, 2009. "Tree structured independence for exponential Brownian functionals," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3798-3815, October.
    2. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2013. "On the exact and approximate distributions of the product of a Wishart matrix with a normal vector," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 70-81.
    3. Bobecka, Konstancja, 2015. "The Matsumoto–Yor property on trees for matrix variates of different dimensions," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 22-34.
    4. Tounsi, Mariem & Zine, Raoudha, 2012. "The inverse Riesz probability distribution on symmetric matrices," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 174-182.
    5. V. Seshadri & J. Wesołowski, 2008. "More on connections between Wishart and matrix GIG distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 68(2), pages 219-232, September.
    6. Wesolowski, Jacek & Witkowski, Piotr, 2007. "Hitting times of Brownian motion and the Matsumoto-Yor property on trees," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1303-1315, September.
    7. Piliszek, Agnieszka & Wesołowski, Jacek, 2016. "Kummer and gamma laws through independences on trees—Another parallel with the Matsumoto–Yor property," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 15-27.
    8. Koudou, Angelo Efoevi, 2012. "A Matsumoto–Yor property for Kummer and Wishart random matrices," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1903-1907.
    9. Kozubowski, Tomasz J. & Mazur, Stepan & Podgórski, Krzysztof, 2022. "Matrix Gamma Distributions and Related Stochastic Processes," Working Papers 2022:12, Örebro University, School of Business.
    10. Letac, Gérard & Wesołowski, Jacek, 2020. "Multivariate reciprocal inverse Gaussian distributions from the Sabot–Tarrès–Zeng integral," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    11. Wesołowski, Jacek, 2015. "On the Matsumoto–Yor type regression characterization of the gamma and Kummer distributions," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 145-149.
    12. Bartosz Kołodziejek, 2017. "The Matsumoto–Yor Property and Its Converse on Symmetric Cones," Journal of Theoretical Probability, Springer, vol. 30(2), pages 624-638, June.

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