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.

中文翻译：

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对称矩阵的扩展矩阵-变量Beta概率分布

使用实对称矩阵空间中的广义幂函数的概念来引入一种扩展的矩阵变量β函数。借助于此，我们定义了扩展的矩阵变量beta分布的不同版本。建立了这些分布的一些基本属性。我们表明，对第一类和第二类扩展矩阵变量β分布使用线性变换，可以概括这些分布。我们还表明，由Tounsi和Zine引入的两个独立的逆Riesz矩阵之和的分布（J Multivar Anal 111：174-182，2012）可以用广义扩展矩阵变量β函数表示。最后，使用定点迭代法，我们根据扩展的矩阵变量beta先验分布的类别为多元正态分布的未知协方差矩阵提供了一个可计算的最大后验（MAP）估计量。此外，我们通过计算诸如均方误差（M S E）和希尔伯特-施密特距离（D H S）。获得的结果证实了提出的先验的性能。