Generalized Bivariate Kummer-Beta Distribution
Main Article Content
Keywords
Beta function, beta distribution, entropy, bivariate distribution, gamma function, Kummer-beta distribution
Abstract
A new bivariate beta distribution based on the Humbert’s confluent hypergeometric function of the second kind is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and entropies.
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References
[1] P. Bran-Cardona, J. Orozco, and D. Nagar, “Generalización bivariada de la distribución kummer-beta,” Revista Colombiana de Estadística, vol. 34, no. 3, pp. 497–512, sep. 2011. https://revistas.unal.edu.co/index.php/estad/
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https://doi.org/10.2307/2348645
[5] T. P. Hutchinson and C. D. Lai, The engineering statistician’s guide to continuous bivariate
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[6] S. Kotz, N. Balakrishnan, and N. Johnson, Continuous Multivariate Distributions, Volume 1: Models and Applications, ser. Continuous Multivariate Distributions. Wiley, 2004.
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[10] S. Nadarajah, “The bivariate f3-beta distribution,” Commun. Korean Math. Soc., vol. 21, no. 2, pp. 363–374, 2006. https://www.koreascience.or.kr/article/JAKO200626813055203.pdf
[11] S. Nadarajah, “The bivariate f2–beta distribution,” American Journal of Mathematical and Management Sciences, vol. 27, no. 3–4, pp. 351–368, 2007. https://doi.org/10.1080/01966324.2007.10737705
[12] N. Saralees and K. Samuel, “The bivariate f1-beta distribution,” C. R. Math. Acad. Sci. Soc. R. Can., vol. 27, no. 2, pp. 58–64, 2005.
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[14] Nadarajah and S. Kotz, “Multitude of beta distributions with applications,” Statistics, vol. 41, no. 2, pp. 153–179, 2007. https://doi.org/10.1080/02331880701223522
[15] S. Nadarajah, S. H. Shih, and D. K. Nagar, “A new bivariate beta distribution,” Statistics, vol. 51, no. 2, pp. 455–474, 2017. https://doi.org/10.1080/02331888.2016.1240681
[16] D. Nagar, S. Nadarajah, and I. Okorie, “A new bivariate distribution with one marginal defined on the unit interval,” Annals of Data Science, vol. 4, no. 3, pp. 405–420, 2017. https://doi.org/10.1007/s40745-017-0111-6
[17] O. Johanna Marcela, N. Daya K., and A. K. Gupta, “Generalized bivariate beta distributions involving appell’s hypergeometric function of the second kind,” Computers & Mathematics with Applications, vol. 64, no. 8, pp. 2507– 2519, 2012. https://doi.org/10.1016/j.camwa.2012.06.006
[18] J. M. Sarabia and E. Castillo, Bivariate Distributions Based on the Generalized Three-Parameter Beta Distribution. Boston, MA: Birkhäuser Boston, 2006, pp. 85–110. https://doi.org/10.1007/0-8176-4487-3_6
[19] A. Gupta and D. Nagar, Matrix Variate Distributions, ser. Monographs and Surveys in Pure and Applied Mathematics. CRC Press, 2018.
[20] A. K. Gupta, C. Liliam, and D. K. Nagar, “variate kummer-dirichlet vistributions,” Journal of Applied Mathematics, vol. 1, no. 3, pp. 117–139, 2001. https://doi.org/10.1155/S1110757X0100701X
[21] D. K. Nagar and A. K. Gupta, “Matrix-variate kummer-beta distribution,” Journal of the Australian Mathematical Society, vol. 73, no. 1, pp. 11 – 26, 2002. https://doi.org/10.1017/S1446788700008442
[22] E. L. Lehmann, “Some concepts of dependence,” The Annals of Mathematical Statistics, vol. 37, no. 5, pp. 1137–1153, 1966. http://www.jstor.org/stable/2239070
[23] Y. L. Tong, Probability Inequalities in Multivariate Distributions, ser. Probability and mathematical statistics. Elsevier Science, 2014.
[24] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, July 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
[25] A. Rényi, “On measures of entropy and information,” Proc. 4th Berkeley Sympos. Math. Statist. and Prob., vol. 61, no. 1, pp. 547–561, 1961. https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s4_v1_article-27.pdf
[26] K. Zografos and S. Nadarajah, “Expressions for rényi and shannon entropies for multivariate distributions,” Statistics & Probability Letters, vol. 71, no. 1, pp. 71 – 84, 2005. https://doi.org/10.1016/j.spl.2004.10.023
[27] K. Zografos and S. Nadarajah, “Expressions for rényi and shannon entropies for multivariate distributions,” Statistics & Probability Letters, vol. 71, no. 1, pp. 71–84, 2005. https://doi.org/10.1016/j.spl.2004.10.023
[28] K. Zografos, “On maximum entropy characterization of pearson’s type ii and vii multivariate distributions,” Journal of Multivariate Analysis, vol. 71, no. 1, pp. 67 – 75, 1999. https://doi.org/10.1006/jmva.1999.1824
[29] D. Nagar and Z. Edwin, “Distributions of the product and the quotient of independent kummer-beta variables,” Scientiae Mathematicae Japonicae, vol. 61, no. 1, pp. 109–117, 2005.
[30] H. Srivastava and P. Karlsson, Multiple Gaussian Hypergeometric Series, ser. Ellis Horwood series in mathematics and its applications. E. Horwood, 1985.
[31] Y. Luke, The Special Functions and Their Approximations. Academic Press, New York, 1969, vol. I.
[32] K. W. NG, “Kummer-gamma and kummer-beta univariate and multivariate distributions,” 1995. https://ci.nii.ac.jp/naid/10015391335/en/
article/view/29965
[2] B. Arnold, E. Castillo, J. Sarabia, J. Sarabia, and J. Sarabia, Conditional Specification of Statistical Models, ser. Springer Series in Statistics. Springer, 1999.
[3] N. Balakrishnan and C. Lai, Continuous Bivariate Distributions. Springer New York, 2009.
[4] T. P. Hutchinson and C. D. Lai, “Reviewed work: Continuous bivariate distributions, emphasising applications,” Journal of the Royal Statistical Society. Series D (The Statistician), vol. 41, no. 1, pp. 125–127, 1992.
https://doi.org/10.2307/2348645
[5] T. P. Hutchinson and C. D. Lai, The engineering statistician’s guide to continuous bivariate
distributions, ser. Adelaide. Rumsby Scientific Publishing, 1991.
[6] S. Kotz, N. Balakrishnan, and N. Johnson, Continuous Multivariate Distributions, Volume 1: Models and Applications, ser. Continuous Multivariate Distributions. Wiley, 2004.
[7] K. V. Mardia, Families of bivariate distributions, ser. Griffin’s Statistical Monographs and Courses, No. 27. Lubrecht & Cramer Ltd, 1970.
[8] D. I. Ghosh, “On the reliability for some bivariate dependent beta and kumaraswamy distributions: A brief survey,” Stochastics and Quality Control, vol. 34, no. 2, pp. 115–121, 2019. https://doi.org/10.1515/eqc-2018-0029
[9] A. K. Gupta, J. M. Orozco, and D. K. Nagar, “Non-central bivariate beta distribution,” Stat. Papers, vol. 52, no. 1, pp. 139–152, 2011. https://doi.org/10.1007/s00362-009-0215-y
[10] S. Nadarajah, “The bivariate f3-beta distribution,” Commun. Korean Math. Soc., vol. 21, no. 2, pp. 363–374, 2006. https://www.koreascience.or.kr/article/JAKO200626813055203.pdf
[11] S. Nadarajah, “The bivariate f2–beta distribution,” American Journal of Mathematical and Management Sciences, vol. 27, no. 3–4, pp. 351–368, 2007. https://doi.org/10.1080/01966324.2007.10737705
[12] N. Saralees and K. Samuel, “The bivariate f1-beta distribution,” C. R. Math. Acad. Sci. Soc. R. Can., vol. 27, no. 2, pp. 58–64, 2005.
[13] S. Nadarajah and S. Kotz, “Some bivariate beta distributions,” Statistics, vol. 39, no. 5, pp. 457–466, 2005. https://doi.org/10.1080/02331880500286902
[14] Nadarajah and S. Kotz, “Multitude of beta distributions with applications,” Statistics, vol. 41, no. 2, pp. 153–179, 2007. https://doi.org/10.1080/02331880701223522
[15] S. Nadarajah, S. H. Shih, and D. K. Nagar, “A new bivariate beta distribution,” Statistics, vol. 51, no. 2, pp. 455–474, 2017. https://doi.org/10.1080/02331888.2016.1240681
[16] D. Nagar, S. Nadarajah, and I. Okorie, “A new bivariate distribution with one marginal defined on the unit interval,” Annals of Data Science, vol. 4, no. 3, pp. 405–420, 2017. https://doi.org/10.1007/s40745-017-0111-6
[17] O. Johanna Marcela, N. Daya K., and A. K. Gupta, “Generalized bivariate beta distributions involving appell’s hypergeometric function of the second kind,” Computers & Mathematics with Applications, vol. 64, no. 8, pp. 2507– 2519, 2012. https://doi.org/10.1016/j.camwa.2012.06.006
[18] J. M. Sarabia and E. Castillo, Bivariate Distributions Based on the Generalized Three-Parameter Beta Distribution. Boston, MA: Birkhäuser Boston, 2006, pp. 85–110. https://doi.org/10.1007/0-8176-4487-3_6
[19] A. Gupta and D. Nagar, Matrix Variate Distributions, ser. Monographs and Surveys in Pure and Applied Mathematics. CRC Press, 2018.
[20] A. K. Gupta, C. Liliam, and D. K. Nagar, “variate kummer-dirichlet vistributions,” Journal of Applied Mathematics, vol. 1, no. 3, pp. 117–139, 2001. https://doi.org/10.1155/S1110757X0100701X
[21] D. K. Nagar and A. K. Gupta, “Matrix-variate kummer-beta distribution,” Journal of the Australian Mathematical Society, vol. 73, no. 1, pp. 11 – 26, 2002. https://doi.org/10.1017/S1446788700008442
[22] E. L. Lehmann, “Some concepts of dependence,” The Annals of Mathematical Statistics, vol. 37, no. 5, pp. 1137–1153, 1966. http://www.jstor.org/stable/2239070
[23] Y. L. Tong, Probability Inequalities in Multivariate Distributions, ser. Probability and mathematical statistics. Elsevier Science, 2014.
[24] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, July 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
[25] A. Rényi, “On measures of entropy and information,” Proc. 4th Berkeley Sympos. Math. Statist. and Prob., vol. 61, no. 1, pp. 547–561, 1961. https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s4_v1_article-27.pdf
[26] K. Zografos and S. Nadarajah, “Expressions for rényi and shannon entropies for multivariate distributions,” Statistics & Probability Letters, vol. 71, no. 1, pp. 71 – 84, 2005. https://doi.org/10.1016/j.spl.2004.10.023
[27] K. Zografos and S. Nadarajah, “Expressions for rényi and shannon entropies for multivariate distributions,” Statistics & Probability Letters, vol. 71, no. 1, pp. 71–84, 2005. https://doi.org/10.1016/j.spl.2004.10.023
[28] K. Zografos, “On maximum entropy characterization of pearson’s type ii and vii multivariate distributions,” Journal of Multivariate Analysis, vol. 71, no. 1, pp. 67 – 75, 1999. https://doi.org/10.1006/jmva.1999.1824
[29] D. Nagar and Z. Edwin, “Distributions of the product and the quotient of independent kummer-beta variables,” Scientiae Mathematicae Japonicae, vol. 61, no. 1, pp. 109–117, 2005.
[30] H. Srivastava and P. Karlsson, Multiple Gaussian Hypergeometric Series, ser. Ellis Horwood series in mathematics and its applications. E. Horwood, 1985.
[31] Y. Luke, The Special Functions and Their Approximations. Academic Press, New York, 1969, vol. I.
[32] K. W. NG, “Kummer-gamma and kummer-beta univariate and multivariate distributions,” 1995. https://ci.nii.ac.jp/naid/10015391335/en/
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Published
Nov 11, 2020
Article Details
How to Cite
Nagar, D. K., Zarrazola, E., & Serna-Morales, J. (2020). Generalized Bivariate Kummer-Beta Distribution. Ingeniería Y Ciencia, 16(32), 7–31. https://doi.org/10.17230/ingciencia.16.32.1
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Universidad de Antioquia
Author Biographies
Daya K. Nagar, Universidad de Antioquia
PhD in Science, profesor titular, Departamento de Matemáticas, Universidad de Antioquia, Medellín, Colombia.
Edwin Zarrazola, Universidad de Antioquia
Magíster en matemáticas, profesor asistente, Departamento de Matemáticas, Universidad de Antioquia, Medellín, Colombia.
Jessica Serna-Morales, Universidad Nacional de Colombia
Magíster en Estadística. Universidad Nacional de Colombia, sede Medellín
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