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Erdélyi-Kober fractional integral operators from a statistical perspective. II. (English) Zbl 1426.45006

Summary: In this paper we examine the densities of a product and a ratio of two real positive definite matrix-variate random variables \(X_1\) and \(X_2\), which are statistically independently distributed, and we consider the density of the product \(U_1=X_2^{\frac{1}{2}}X_1X_2^{\frac{1}{2}}\) as well as the density of the ratio \(U_1=X_2^{\frac{1}{2}}X_1^{-1}X_2^{\frac{1}{2}}\). We define matrix-variate Kober fractional integral operators of the first and second kinds from a statistical perspective, making use of the derivation in the predecessor of this paper for the scalar variable case, by deriving the densities of product cand ratios where one variable has a matrix-variate type-1 beta density and the other matrix variable has an arbitrary density, in the sense, any real-valued scalar function \(f(X)\) of matrix argument \(X\), such that \(f(X)\) is non-negative for all \(X\) and the total integral over all \(X\), on the support of \(f(X)\), is unity. A number of generalizations are considered, by using pathway models, by appending matrix variate hypergeometric series etc. During this process matrix-variate Saigo operator and other operators are also defined and properties studied.
For Part I see [the authors, Tbil. Math. J. 10, No. 1, 145–159 (2017; Zbl 1360.45011)].

MSC:

45P05 Integral operators
26A33 Fractional derivatives and integrals
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
62H05 Characterization and structure theory for multivariate probability distributions; copulas

Citations:

Zbl 1360.45011
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References:

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