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On Riesz and Wishart distributions associated with decomposable undirected graphs

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  • Andersson, Steen A.
  • Klein, Thomas

Abstract

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4] and [8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.

Suggested Citation

  • Andersson, Steen A. & Klein, Thomas, 2010. "On Riesz and Wishart distributions associated with decomposable undirected graphs," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 789-810, April.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:4:p:789-810
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    References listed on IDEAS

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    1. Steen A. Andersson & David Madigan & Michael D. Perlman, 2001. "Alternative Markov Properties for Chain Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 33-85, March.
    2. Andersson, Steen A. & Perlman, Michael D., 1998. "Normal Linear Regression Models With Recursive Graphical Markov Structure," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 133-187, August.
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    Cited by:

    1. Abdelhamid Hassairi & Fatma Ktari & Raoudha Zine, 2022. "On the Gaussian representation of the Riesz probability distribution on symmetric matrices," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(4), pages 609-632, December.
    2. Andre Lucas & Anne Opschoor & Luca Rossini, 2021. "Tail Heterogeneity for Dynamic Covariance Matrices: the F-Riesz Distribution," Tinbergen Institute Discussion Papers 21-010/III, Tinbergen Institute, revised 11 Jul 2023.
    3. Kammoun, Kaouthar & Louati, Mahdi & Masmoudi, Afif, 2017. "Maximum likelihood estimator of the scale parameter for the Riesz distribution," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 127-131.
    4. Emna Ghorbel & Mahdi Louati, 2024. "An expectation maximization algorithm for the hidden markov models with multiparameter student-t observations," Computational Statistics, Springer, vol. 39(6), pages 3287-3301, September.
    5. Louati, Mahdi & Masmoudi, Afif, 2015. "Moment for the inverse Riesz distributions," Statistics & Probability Letters, Elsevier, vol. 102(C), pages 30-37.
    6. Gribisch, Bastian & Hartkopf, Jan Patrick, 2023. "Modeling realized covariance measures with heterogeneous liquidity: A generalized matrix-variate Wishart state-space model," Journal of Econometrics, Elsevier, vol. 235(1), pages 43-64.
    7. Piotr Graczyk & Hideyuki Ishi & Salha Mamane, 2019. "Wishart exponential families on cones related to tridiagonal matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(2), pages 439-471, April.

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