Jones-Balakrishnan property for matrix variate beta distributions. (English) Zbl 1519.60024
Summary: Let \(\boldsymbol{X}\) and \(\boldsymbol{Y}\) be independent \(m \times m\) symmetric positive definite random matrices. Assume that \(\boldsymbol{X}\) follows a matrix variate beta distribution with parameters \(a\) and \(b\) and that \(\boldsymbol{Y}\) 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 \(\boldsymbol{I}_m\) is an identity matrix and \(\boldsymbol{A}^{1/2}\) is the unique symmetric positive definite square root of \(\boldsymbol{A}\). In this paper, we have shown that random matrices \(\boldsymbol{R}\) and \(\boldsymbol{S}\) are independent and follow matrix variate beta distributions generalizing an independence property established by M. C. Jones and N. Balakrishnan [Stat. Probab. Lett. 170, Article ID 109011, 6 p. (2021; Zbl 1457.60024)], in the univariate case.
MSC:
| 60E05 | Probability distributions: general theory |
| 62E10 | Characterization and structure theory of statistical distributions |
| 62H10 | Multivariate distribution of statistics |
Citations:
Zbl 1457.60024References:
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