Jones-Balakrishnan Property for Matrix Variate Beta Distributions

Daya K. Nagar, Alejandro Roldán-Correa and Saralees Nadarajah ()
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Daya K. Nagar: Universidad de Antioquia
Alejandro Roldán-Correa: Universidad de Antioquia
Saralees Nadarajah: University of Manchester

Sankhya A: The Indian Journal of Statistics, 2023, vol. 85, issue 2, No 15, 1489-1509

Abstract: 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 R = I m − Y + Y 1 / 2 X Y 1 / 2 − 1 / 2 Y 1 / 2 X Y 1 / 2 $\boldsymbol {R}= \left (\boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{-1/2} \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}$ I m − Y + Y 1 / 2 X Y 1 / 2 − 1 / 2 $ \left (\boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}\right )^{-1/2} $ and S = I m − Y + Y 1 / 2 X Y 1 / 2 $\boldsymbol {S}= \boldsymbol {I}_{m} - \boldsymbol {Y} + \boldsymbol {Y}^{1/2} \boldsymbol {X} \boldsymbol {Y}^{1/2}$ , 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.

Keywords: Beta distribution; independence; matrix valued function; transformation.; 60E05; 62E10; 62H10 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13171-022-00299-y

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