1Universidad del Valle, Facultad de Ciencias, Departamento de Matemáticas, Cali, Colombia. Lecturer. Email: paula.bran@gmail.com
2Universidad de Antioquía, Facultad de Ciencias Naturales y Exactas, Departamento de Matemáticas, Medellín, Colombia. Lecturer. Email: jmoc03@gmail.com
3Universidad de Antioquía, Facultad de Ciencias Naturales y Exactas, Departamento de Matemáticas, Medellín, Colombia. Professor. Email: dayaknagar@yahoo.com ]]>
In this article, we study several properties such as marginal and conditional distributions, joint moments, and mixture representation of the bivariate generalization of the Kummer-Beta distribution. To show the behavior of the density function, we give some graphs of the density for different values of the parameters. Finally, we derive the exact and approximate distribution of the product of two random variables which are distributed jointly as bivariate Kummer-Beta. The exact distribution of the product is derived as an infinite series involving Gauss hypergeometric function, whereas the beta distribution has been used as an approximate distribution. Further, to show the closeness of the approximation, we have compared the exact distribution and the approximate distribution by using several graphs. An application of the results derived in this article is provided to visibility data from Colombia.
Key words: Beta distribution, Bivariate distribution, Dirichlet distribution, Hypergeometric function, Moments, Transformation.
En este artículo, definimos la función de densidad de la generalización bivariada de la distribución Kummer-Beta. Estudiamos algunas de sus propiedades y casos particulares, así como las distribuciones marginales y condicionales. Para ilustrar el comportamiento de la función de densidad, mostramos algunos gráficos para diferentes valores de los parámetros. Finalmente, encontramos la distribución del producto de dos variables cuya distribución conjunta es Kummer-Beta bivariada y utilizamos la distribución beta como una aproximación. Además, con el fin de comparar la distribución exacta y la aproximada de este producto, mostramos algunos gráficos. Se presenta una aplicación a datos climáticos sobre niebla y neblina de Colombia.
Palabras clave: distribución Beta, distribución bivariada, distribución Dirichlet, función hipergeométrica, momentos, transformación.
Texto completo disponible en PDF
1. Balakrishnan, N. & Lai, C. D. (2009), Continuous Bivariate Distributions, second edn, Springer. [ Links ]
2. Barry, C. A., Castillo, E. & Sarabia, J. M. (1999), Conditional Specification of Statistical Models, Springer Series in Statistics, Springer-Verlag, New York. [ Links ]
3. Gordy, M. (1998), 'Computationally convenient distributional assumptions for common-value auctions', Computational Economics 12, 61-78. [ Links ]
4. Gupta, A. K., Carde\~no, L. & Nagar, D. K. (2001), 'Matrix variate Kummer-Dirichlet distributions', Journal of Applied Mathematics 1(3), 117-139. [ Links ]
5. Gupta, A. K. & Nagar, D. K. (2000), Matrix Variate Distributions, Vol. 104 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL. [ Links ]
6. Gupta, A. K. & Song, D. (1996), 'Generalized Liouville distribution', Computers & Mathematics with Applications 32(2), 103-109. [ Links ]
7. Gupta, A. K. & Wong, C. F. (1985), 'On three and five parameter bivariate Beta distributions', International Journal for Theoretical and Applied Statistics 32(2), 85-91. [ Links ]
8. Gupta, R. D. & Richards, D. S. P. (2001), 'The history of the Dirichlet and Liouville distributions', International Statistical Review 69(3), 433-446. [ Links ]
9. Hahn, G. J. & Shapiro, S. S. (1967), Statistical Models in Engineering, John Wiley and Sons, New York. [ Links ]
10. Hutchinson, T. P. & Lai, C. D. (1991), The Engineering Statistician's Guide To Continuous Bivariate Distributions, Rumsby Scientific Publishing, Adelaide. [ Links ]
11. Kotz, S., Balakrishnan, N. & Johnson, N. L. (2000), Continuous Multivariate Distributions. Vol. 1. Models and applications, Wiley Series in Probability and Statistics: Applied Probability and Statistics, Second edn, Wiley-Interscience, New York. [ Links ]
12. Lehmann, E. L. (1966), 'Some concepts of dependence', Annals of Mathematical Statistics 37, 1137-1153. [ Links ]
13. Luke, Y. L. (1969), The Special Functions and their Approximations, Vol. 53 of Mathematics in Science and Engineering, Academic Press, New York. [ Links ]
14. Mardia, K. V. (1970), Families of Bivariate Distributions, Hafner Publishing Co., Darien, Conn.. Griffin's Statistical Monographs and Courses, No. 27. [ Links ]
15. Marshall, A. W. & Olkin, I. (1979), Inequalities: Theory of Majorization and its Applications, Vol. 143 of Mathematics in Science and Engineering, Academic Press Inc., New York. [ Links ]
16. Nadarajah, S. & Kotz, S. (2005), 'Some bivariate Beta distributions', A Journal of Theoretical and Applied Statistics 39(5), 457-466. [ Links ]
17. Nadarajah, S. & Zografos, K. (2005), 'Expressions for Rényi and Shannon entropies for bivariate distributions', Information Sciences 170(2-4), 173-189. *http://dx.doi.org/10.1016/j.ins.2004.02.020 [ Links ]
18. Nagar, D. K. & Gupta, A. K. (2002), 'Matrix-variate Kummer-Beta distribution', Journal of the Australian Mathematical Society 73(1), 11-25. [ Links ]
19. Nagar, D. K. & Zarrazola, E. (2005), 'Distributions of the product and the quotient of independent Kummer-Beta variables', Scientiae Mathematicae Japonicae 61(1), 109-117. [ Links ]
20. Ng, K. W. & Kotz, S. (1995), Kummer-Gamma and Kummer-Beta univariate and multivariate distributions, 84, Department of Statistics, The University of Hong Kong, Hong Kong %month 01. [ Links ]
]]>21. Rényi, Alfréd (1961), On measures of entropy and information, 'Procedings 4th Berkeley Symposium Mathematical Statistics and Probability', University of California Press, Berkeley, California, p. 547-561. [ Links ]
22. Shannon, C. E. (1948), 'A mathematical theory of communication', The Bell System Technical Journal 27, 379-423, 623-656. [ Links ]
23. Sivazlian, B. D. (1981), 'On a multivariate extension of the Gamma and Beta distributions', SIAM Journal on Applied Mathematics 41(2), 205-209. [ Links ]
24. Song, D. & Gupta, A. K. (1997), 'Properties of generalized Liouville distributions', Random Operators and Stochastic Equations 5(4), 337-348. [ Links ]
25. Zografos, K. (1999), 'On maximum entropy characterization of Pearson's type II and VII multivariate distributions', Journal of Multivariate Analysis 71(1), 67-75. *http://dx.doi.org/10.1006/jmva.1999.1824 [ Links ]
26. Zografos, K. & Nadarajah, S. (2005), 'Expressions for Rényi and Shannon entropies for multivariate distributions', Statistics & Probability Letters 71(1), 71-84. *http://dx.doi.org/10.1016/j.spl.2004.10.023 [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv34n3a07,
AUTHOR = {Bran-Cardona, Paula Andrea and Orozco-Casta\~neda, Johanna Marcela and Nagar, Daya Krishna},
TITLE = {{Bivariate Generalization of the Kummer-Beta Distribution}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2011},
volume = {34},
number = {3},
pages = {497-512}
}