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Bivariate Distributions Based on the Generalized Three-Parameter Beta Distribution

  • José María Sarabia
  • Enrique Castillo
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

The generalized three-parameter beta distribution with pdf proportional to x a−1(1−x)b−1/{1−(1−λ)x}a+b is a flexible extension of the classical beta distribution with interesting applications in statistics. In this chapter, several bivariate extensions of this distribution are studied. We propose models with given marginals: a first model consists of a transformation with monotonic components of the Dirichlet distribution and a second model that uses the bivariate Sarmanov-Lee distribution. Next, the class of distributions whose conditionals belong to the generalized three-parameter beta distribution is considered. Two important subfamilies are studied in detail. The first one contains as a particular case the models of (1982) and (2003). The second family is more general, and contains among others, the model proposed by (1999). In addition, using two different conditional schemes, we study conditional survival models. Multivariate extensions are also discussed. Finally, an application to Bayesian analysis is given.

Keywords and phrases:

Generalized three-parameter beta distribution Gauss hypergeometric distribution Dirichlet and Sarmanov-Lee distributions conditionally specified models 

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References

  1. 1.
    Armero, C., and Bayarri, M. J. (1994). Prior assessments for prediction in queues, Journal of the Royal Statistical Society, Series D, 43, 139–153.Google Scholar
  2. 2.
    Arnold, B. C. (1992). Conditional survival models, In Recent Advances in Life-Testing and Reliability: A Volume in Honor of Alonzo Clifford Cohen Jr. (Ed., N. Balakrishnan), pp. 589–601, CRC Press, Boca Raton.Google Scholar
  3. 3.
    Arnold, B. C., Castillo, E. and Sarabia, J. M. (1992). Conditionally Specified Distributions, Lecture Notes in Statistics, Vol. 73, Springer-Verlag, New York.zbMATHGoogle Scholar
  4. 4.
    Arnold, B. C., Castillo, E., and Sarabia, J. M. (1998). Bayesian analysis for classical distributions using conditionally specified prior, Sankhyā, Series B, 60, 228–245.zbMATHMathSciNetGoogle Scholar
  5. 5.
    Arnold, B. C., Castillo, E., and Sarabia, J. M. (1999). Conditional Specification of Statistical Models, Springer-Verlag, New York.zbMATHGoogle Scholar
  6. 6.
    Arnold, B. C., Castillo, E., and Sarabia, J. M. (2001). Conditionally specified distributions: An introduction (with discussion), Statistical Science, 16, 249–274.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Arnold, B. C., Castillo, E., Sarabia, J. M., and González-Vega, L. (2000). Multiple modes in densities with normal conditionals, Statistics and Probability Letters, 49, 355–363.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Arnold, B. C., and Strauss, D. (1991). Bivariate distributions with conditionals in prescribed exponential families, Journal of the Royal Statistical Society, Series B, 53, 365–375.zbMATHMathSciNetGoogle Scholar
  9. 9.
    Balakrishnan, N. (Ed.) (1992). Handbook of Logistic Distribution, Marcel Dekker, New York.zbMATHGoogle Scholar
  10. 10.
    Barlow, R. E., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing: Probability Models, To Begin With, Silver Springs, Maryland.Google Scholar
  11. 11.
    Castillo, E., and Sarabia, J. M. (1990). Bivariate distributions with second kind beta conditionals Communications in Statistics—Theory and Methods, 19, 3433–3445.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen, J. J., and Novick, M. R. (1984). Bayesian analysis for binomial models with generalized beta prior distributions, Journal of Educational Statistics, 9, 163–175.CrossRefGoogle Scholar
  13. 13.
    Fisher, R. A. (1924). On a distribution yielding the error functions of several well known statistics, In Proceedings of the International Mathematical Congress, Toronto, Vol. 2, pp. 805–813.Google Scholar
  14. 14.
    Holland, P. W., and Wang, Y. L. (1987). Dependence function for continuous bivariate densities, Communications in Statistics—Theory and Methods, 16, 863–876.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Johnson, N. L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2, Second edition, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  16. 16.
    Jones, M. C. (1996). The local dependence function, Biometrika, 83, 899–904.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jones, M. C. (2004). Families of distributions arising from distributions of order statistics, Test, 13, 1–43.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 2, Second edition, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  19. 19.
    Lee, M.-L. T. (1996). Properties and applications of the Sarmanov family of bivariate distributions, Communications in Statistics—Theory and Methods, 25, 1207–1222.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Libby, D. L., and Novick, M. R. (1982). Multivariate generalized beta distributions with applications to utility assessment, Journal of Educational Statistics, 7, 271–294.CrossRefGoogle Scholar
  21. 21.
    Olkin, I. and Liu, R. (2003). A bivariate beta distribution, Statistics and Probability Letters, 62, 407–412.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pham-Gia, T., and Duong, Q. P (1989). The generalized beta and F-distributions in statistical modelling, Mathematical and Computer Modelling, 12, 1613–1625.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sarmanov, O. V. (1966). Generalized normal correlation and two-dimensional Frechet classes, Doklady, Soviet Mathematics, 168, 596–599.MathSciNetGoogle Scholar
  24. 24.
    Stuart, A., and Ord, J. K. (1987). Kendall’s Advanced Theory of Statistics—Vol. 1: Distribution Theory, Oxford University Press, New York.Google Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • José María Sarabia
    • 1
  • Enrique Castillo
    • 1
  1. 1.University of CantabriaSantanderSpain

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