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The Extended Matrix-Variate Beta Probability Distribution on Symmetric Matrices

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  • Mariem Tounsi

    (Sfax University)

Abstract

The notion of generalized power function in the space of real symmetric matrices is used to introduce a kind of extended matrix-variate beta function. With the aid of this, we define a different versions of extended matrix-variate beta distributions. Some fundamental properties of these distributions are established. We show that using a linear transformation on the extended matrix-variate beta distributions of the first and second kind, we can generalize these distributions. We also show that the distribution of the sum of two independent inverse Riesz matrices introduced by Tounsi and Zine (J Multivar Anal 111:174–182, 2012) can be written in terms of the generalized extended matrix-variate beta function. Finally, using Fixed point iterative method, we provide a calculable maximum a posteriori (MAP) estimator for the unknown covariance matrix of a multivariate normal distribution based on the class of the extended matrix-variate beta prior distribution. Additionally, we evaluated the Gaussian finite sample performance by calculating such evaluation criteria as Mean Square Error (MSE) and Hilbert-Schmidt distance (DHS). The obtained results confirm the performance of the proposed prior.

Suggested Citation

  • Mariem Tounsi, 2020. "The Extended Matrix-Variate Beta Probability Distribution on Symmetric Matrices," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 647-676, June.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:2:d:10.1007_s11009-019-09725-5
    DOI: 10.1007/s11009-019-09725-5
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    References listed on IDEAS

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    1. McDonald, James B. & Xu, Yexiao J., 1995. "A generalization of the beta distribution with applications," Journal of Econometrics, Elsevier, vol. 69(2), pages 427-428, October.
    2. Alexander, Carol & Cordeiro, Gauss M. & Ortega, Edwin M.M. & Sarabia, José María, 2012. "Generalized beta-generated distributions," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1880-1897.
    3. Gordy, Michael B, 1998. "Computationally Convenient Distributional Assumptions for Common-Value Auctions," Computational Economics, Springer;Society for Computational Economics, vol. 12(1), pages 61-78, August.
    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. Simsek, Yilmaz, 2015. "Beta-type polynomials and their generating functions," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 172-182.
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    Cited by:

    1. Shokofeh Zinodiny & Saralees Nadarajah, 2022. "Matrix Variate Two-Sided Power Distribution," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 179-194, March.

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