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Jones-Balakrishnan Property for Matrix Variate Beta Distributions

Author

Listed:
  • Daya K. Nagar

    (Universidad de Antioquia)

  • Alejandro Roldán-Correa

    (Universidad de Antioquia)

  • Saralees Nadarajah

    (University of Manchester)

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.

Suggested Citation

  • Daya K. Nagar & Alejandro Roldán-Correa & Saralees Nadarajah, 2023. "Jones-Balakrishnan Property for Matrix Variate Beta Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1489-1509, August.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:2:d:10.1007_s13171-022-00299-y
    DOI: 10.1007/s13171-022-00299-y
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    References listed on IDEAS

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    1. Jones, M.C. & Balakrishnan, N., 2021. "Simple functions of independent beta random variables that follow beta distributions," Statistics & Probability Letters, Elsevier, vol. 170(C).
    2. D. K. Nagar & M. Arashi & S. Nadarajah, 2017. "Bimatrix variate gamma-beta distributions," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(9), pages 4464-4483, May.
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