The Extended Matrix-Variate Beta Probability Distribution on Symmetric Matrices

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.

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Correspondence to Mariem Tounsi.

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Appendix A: The MAP-Fixed Point Algorithm

Appendix A: The MAP-Fixed Point Algorithm

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Tounsi, M. The Extended Matrix-Variate Beta Probability Distribution on Symmetric Matrices. Methodol Comput Appl Probab 22, 647–676 (2020). https://doi.org/10.1007/s11009-019-09725-5

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Keywords

  • Random matrices
  • Extended beta function
  • Division algorithm
  • Inverse Riesz distribution
  • Beta-Riesz distribution
  • Covariance estimation
  • MAP estimator
  • Fixed point algorithm

Mathematics Subject Classification (2010)

  • 62E10
  • 60E05
  • 15A52
  • 15B48