Abstract
Let X and Y be independent m × m symmetric positive definite random matrices. Assume that X follows a matrix variate beta distribution with parameters a and b and that Y has a matrix variate beta distribution with parameters a + b and c. Define \(\boldsymbol {R}= \left (\boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{-1/2} \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\) \( \left (\boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{-1/2} \) and \(\boldsymbol {S}= \boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\), where Im is an identity matrix and A1/2 is the unique symmetric positive definite square root of A. In this paper, we have shown that random matrices R and S are independent and follow matrix variate beta distributions generalizing an independence property established by Jones and Balakrishnan (Statistics and Probability Letters, 170 (2021), article id 109011) in the univariate case.
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Acknowledgments
The research work of Daya K. Nagar was supported by the Sistema Universitario de Investigación, Universidad de Antioquia [project no. 2019-25374]. All of the authors would like to thank the Editor and the referee for careful reading and comments which greatly improved the paper.
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Nagar, D.K., Roldán-Correa, A. & Nadarajah, S. Jones-Balakrishnan Property for Matrix Variate Beta Distributions. Sankhya A 85, 1489–1509 (2023). https://doi.org/10.1007/s13171-022-00299-y
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DOI: https://doi.org/10.1007/s13171-022-00299-y