Advertisement

Distributions of powers of the central beta matrix variates and applications

  • Thu Pham-GiaEmail author
  • Duong Thanh Phong
  • Dinh Ngoc Thanh
Original Paper
  • 15 Downloads

Abstract

We consider the central Beta matrix variates of both kinds, and establish the expressions of the densities of integral powers of these variates, for all their three types of distributions encountered in the statistical literature: entries, determinant, and latent roots distributions. Applications and computation of credible intervals are presented.

Keywords

Beta matrix variates Credible interval G-Function Latent roots Powers 

Mathematics Subject Classification

62H10 

Notes

Acknowledgements

The authors wish to thank two anonymous referees for their constructive criticisms and suggestions that have helped them to improve the quality of their paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Bekker A, Roux JJ, Pham-Gia T (2009) The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables. Stat Pap 50(2):323–338MathSciNetCrossRefGoogle Scholar
  2. Gupta AK, Nagar DK (2000) Matrix variate distributions. Chapman and Hall/CRC, New YorkzbMATHGoogle Scholar
  3. Lawley D (1938) A generalization of Fisher’s \(z\) test. Biometrika 30(1/2):180–187CrossRefGoogle Scholar
  4. Mathai AM (1984) Extensions of Wilks’ integral equations and distributions of test statistics. Ann Inst Stat Math 36(2):271–288MathSciNetCrossRefGoogle Scholar
  5. Mathai AM (1997) Jacobians of matrix transformations and functions of matrix arguments. World Scientific Publishing Company, SingaporeCrossRefGoogle Scholar
  6. Mathai AM, Saxena R, Haubold H (2010) The \(H\)-function: theory and applications. Springer, New YorkzbMATHGoogle Scholar
  7. McDonald JB, Xu YJ (1995) A generalization of the beta distribution with applications. J Econom 66(1–2):133–152CrossRefGoogle Scholar
  8. Muirhead RJ (1982) Aspects of multivariate statistical theory. Wiley, New YorkCrossRefGoogle Scholar
  9. Nadarajah S, Kotz S (2004) A generalized beta distribution. Math Sci 29:36–41MathSciNetzbMATHGoogle Scholar
  10. Pauw J, Bekker A, Roux JJ (2010) Densities of composite Weibullized generalized gamma variables: theory and methods. South Afr Stat J 44(1):17–42zbMATHGoogle Scholar
  11. Pham-Gia T (2008) Exact distribution of the generalized Wilks’s statistic and applications. J Multivar Anal 99(8):1698–1716MathSciNetCrossRefGoogle Scholar
  12. Pham-Gia T, Thanh DN (2016) Hypergeometric functions: From one scalar variable to several matrix arguments, in statistics and beyond. Open J Stat 6(5):951–994CrossRefGoogle Scholar
  13. Pham-Gia T, Turkkan N (2011a) Distributions of ratios: from random variables to random matrices. Open J Stat 1(2):93–104MathSciNetCrossRefGoogle Scholar
  14. Pham-Gia T, Turkkan N (2011b) Testing the equality of several covariance matrices. J Stat Comput Simul 81(10):1233–1246MathSciNetCrossRefGoogle Scholar
  15. Pillai K (1954) On some distribution problems in multivariate analysis. Tech. Rep., North Carolina State University. Dept. of StatisticsGoogle Scholar
  16. Pillai K (1955) Some new test criteria in multivariate analysis. Ann Math Stat 26(1):117–121MathSciNetCrossRefGoogle Scholar
  17. Rencher A, Christensen W (2012) Methods of multivariate analysis, 3rd edn. Wiley, New YorkCrossRefGoogle Scholar
  18. Roy SN (1953) On a heuristic method of test construction and its use in multivariate analysis. Ann Math Stat 24(2):220–238MathSciNetCrossRefGoogle Scholar
  19. Turkkan N, Pham-Gia T (1993) Computation of the highest posterior density interval in Bayesian analysis. J Stat Comput Simul 44(3–4):243–250CrossRefGoogle Scholar
  20. Turkkan N, Pham-Gia T (1997) Algorithm as 308: highest posterior density credible region and minimum area confidence region: the bivariate case. J R Stat Soc Ser C (Appl Stat) 46(1):131–140CrossRefGoogle Scholar
  21. Wilks S (1932) Certain generalizations in the analysis of variance. Biometrika 24(3/4):471–494CrossRefGoogle Scholar
  22. Wishart J (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20(1/2):32–52CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Thu Pham-Gia
    • 1
    • 4
    Email author
  • Duong Thanh Phong
    • 2
    • 4
  • Dinh Ngoc Thanh
    • 3
    • 4
  1. 1.Département de Mathématiques et de Statistiques, Faculté des SciencesUniversité de MonctonMonctonCanada
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of Science, Vietnam National University Ho Chi Minh CityHo Chi Minh CityVietnam
  4. 4.Applied Multivariate Statistical Analysis Research GroupHo Chi Minh CityVietnam

Personalised recommendations