Bernoulli

  • Bernoulli
  • Volume 14, Number 3 (2008), 749-763.

Kshirsagar–Tan independence property of beta matrices and related characterizations

Konstancja Bobecka and Jacek Wesołowski

Full-text: Open access

Abstract

A new independence property of univariate beta distributions, related to the results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a characterization of univariate beta laws through this independence property is proved. A related characterization of a family of 2×2 random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation.

Article information

Source
Bernoulli, Volume 14, Number 3 (2008), 749-763.

Dates
First available in Project Euclid: 25 August 2008

Permanent link to this document
https://projecteuclid.org/euclid.bj/1219669628

Digital Object Identifier
doi:10.3150/07-BEJ118

Mathematical Reviews number (MathSciNet)
MR2537810

Zentralblatt MATH identifier
1155.62041

Keywords
beta matrix Dirichlet distribution functional equations independence perpetuities univariate beta distribution

Citation

Bobecka, Konstancja; Wesołowski, Jacek. Kshirsagar–Tan independence property of beta matrices and related characterizations. Bernoulli 14 (2008), no. 3, 749--763. doi:10.3150/07-BEJ118. https://projecteuclid.org/euclid.bj/1219669628


Export citation

References

  • Bobecka, K. and Wesołowski, J. (2007). The Dirichlet distribution and process through neutralities., J. Theor. Probab. 20 295–308.
  • Bobecka, K. and Wesołowski, J. (2002). The Lukacs–Olkin–Rubin theorem without invariance of the “quotient”., Studia Mathematica 152 147–160.
  • Casalis, M. and Letac, G. (1996). The Lukacs–Olkin–Rubin characterization of Wishart distributions on symmetric cones., Ann. Statist. 24 763–786.
  • Chamayou, J.F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices., J. Theor. Probab. 4 3–36.
  • Connor, J.R. and Mosimann, J.E. (1969). Concepts of independence for proportions with a generalization of the Dirichlet distribution., J. Amer. Statist. Assoc. 64 194–206.
  • Fabius, J. (1973). Two characterizations of the Dirichlet distribution., Ann. Statist. 1 583–587.
  • Geiger, D. and Heckerman, D. (1997). A characterization of the Dirichlet distribution through global and local parameter independence., Ann. Statist. 25 1344–1369.
  • Geiger, D. and Heckerman, D. (1998). A characterization of the bivariate Wishart distribution., Probab. Math. Statist. 18 119–131.
  • Geiger, D. and Heckerman, D. (2002). Parameter priors for directed acyclic graphical models and the characterization of several probability distributions., Ann. Statist. 30 1412–1440.
  • Goldie, C.M. and Grübel, R. (1996). Perpetuities with thin tails., Adv. in Appl. Probab. 28 463–480.
  • Gupta, A.K. and Nadarajah, S. (2004)., Handbook of Beta Distribution and Its Applications. New York: Dekker.
  • Gupta, A.K. and Nagar, D.K. (2000)., Matrix Variate Distributions. Boca Raton, FL: Chapman and Hall/CRC.
  • Hassairi, A. and Regaig, O. (2006). Characterizations of the beta distribution on symmetric matrices. Preprint, 1-11.
  • James, I.R. and Mosimann, J.E. (1980). A new characterization of the Dirichlet distribution through neutrality., Ann. Statist. 8 183–189.
  • Kshirsagar, A.M. (1961). The non-central multivariate beta distribution., Ann. Math. Statist. 32 104–111.
  • Kshirsagar, A.M. (1972)., Multivariate Analysis. New York: Dekker.
  • Letac, G. and Massam, H. (2001). The normal quasi-Wishart distribution. In, Algebraic Methods in Statistics and Probability (M.A.G. Viana and D.St.P. Richards, eds.). AMS Contemporary Mathematics 287 231–239.
  • Letac, G. and Wesołowski, J. (2000). An independence property for the product of GIG and gamma laws., Ann. Probab. 28 1371–1383.
  • Massam, H. and Wesołowski, J. (2006). The Matsumoto–Yor property and the structure of the Wishart distribution., J. Multivariate Anal. 97 103–123.
  • Muirhead, R.J. (1982)., Aspects of Multivariate Statistical Theory. New York: Wiley.
  • Tan, W.Y. (1969). Note on the multivariate and generalized multivariate beta distributions., J. Amer. Statist. Assoc. 64 230–241.
  • Olkin, I. and Rubin, H. (1964). Multivariate beta distributions and independence properties of the Wishart distribution., Ann. Math. Statist. 35 261–269.
  • Seshadri, V. and Wesołowski, J. (2003). Constancy of regressions for beta distributions., Sankhyā A 65 284–291.
  • Seshadri, V. and Wesołowski, J. (2007). More on connections between Wishart and matrix GIG distributions., Metrika. DOI: 10.1007/s00184-007-0154-3.
  • Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables., Adv. in Appl. Probab. 11 750–783.