A Bivariate Gamma Distribution Whose Marginals are Finite Mixtures of Gamma Distributions

  • Maryam Rafiei Department of Mathematics and Statistics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
  • Anis Iranmanesh Department of Mathematics and Statistics, Mashhad Branch, Islamic Azad University, Mashhad, Iran;
  • Daya k. Nagar Instituto de Mathematicas, Universidad de Antioquia, Medellin, Colombia.
Keywords: Bivariate Distribution, Beta Distribution, Entropy, Information Matrix, Gamma Distribution, Simulation

Abstract

In this article a new bivariate distribution, whose both the marginals are nite mixture ofgamma distribution has been dened. Several of its properties such moments, correlationcoefficients, measure of skewness, moment generating function, Renyi and Shannon entropieshave been derived. Simulation study have been conducted to evaluate the performance ofmaximum likelihood method.

References

N. Balakrishnan and Chin-Diew Lai, Continuous bivariate distributions. Second edition. Springer, Dordrecht, 2009.

Lennart Bondesson, On univariate and bivariate generalized gamma convolutions, Journal of Statistical Planning and Inference, vol. 139, no. 11, pp. 3759–3765, 2009.

F. Chatelain and J. -Y. Tourneret, Bivariate Gamma Distributions for Multisensor Sar Images, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07, Honolulu, HI, 2007, pp. III-1237-III-1240.

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Elsevier/Academic Press, Amsterdam, 2015.

Arjun K. Gupta and Saralees Nadarajah, Sums, products and ratios for McKay’s bivariate gamma distribution, Mathematical and Computer Modelling, vol. 43, no. 1–2, 185–193, 2006.

A. K. Gupta and D. K. Nagar, Matrix variate distributions, Chapman & Hall/CRC, Boca Raton, 2000.

N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous univariate distributions, vol. 2, Second Edition, John Wiley & Sons, New York, 1994.

S. Kotz, N. Balakrishnan and N.L. Johnson, Continuous multivariate distributions, vol. 1 , Second Edition, John Wiley & Sons, New York, 2000.

E. L. Lehmann, Some concepts of dependence, Annals of Mathematical Statistics, vol. 37, pp. 1137–1153, 1966.

Y. L. Luke, The special functions and their approximations, Volume 1, Academic Press, New York, 1969.

Makoto Maejima and Yohei Ueda, A note on a bivariate gamma distribution, Statistics & Probability Letters, vol. 80, no. 23–24, pp. 1991–1994, 2010.

K.V.Mardia,Families of bivariate distributions, Griffin’s Statistical Monographs and Courses, No.27,Hafner Publishing Co., Darien, Conn., 1970.

Saralees Nadarajah, Reliability for some bivariate gamma distributions, Mathematical Problems in Engineering, vol. 2005, no. 2, pp. 151–163, 2005.

Saralees Nadarajah, The bivariate gamma exponential distribution with application to drought data, Journal of Applied Mathematics and Computing, vol. 24, no. 1-2, pp. 221–230, 2007.

Saralees Nadarajah, A bivariate distribution with gamma and beta marginals with application to drought data, Journal of Applied Statistics, vol. 36, no. 3-4, pp 277–30, 2009.

Saralees Nadarajah and Arjun K. Gupta, Some bivariate gamma distributions, Applied Mathematics Letters, vol. 19, no. 8, 767–774, 2006.

S. Nadarajah and K. Zografos, Expressions for Rényi and Shannon entropies for bivariate distributions, Information Sciences, vol. 170, no. 2-4, pp. 173–189, 2005.

Daya K. Nagar, Edwin Zarrazola and Luz Estela Sánchez, A bivariate distribution whose marginal laws are gamma and Macdonald, International Journal of Mathematical Analysis, vol. 10, no. 10, pp. 455–467, 2016.

Daya K. Nagar, S. Nadarajah and Idika E. 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.

T. Piboongungon, V. Aalo, C. Iskander and G. Efthymoglou, Bivariate generalised gamma distribution with arbitrary fading parameters, Electronics Letters, vol. 41, pp. 709–710, 2005.

A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, edited by J. Neyman, University of California Press, Berkeley, California, pp. 547–561, 1961.

Abdus Saboor, Serge B. Provost and Munir Ahmad, Univariate and bivariate gamma-type distributions, Lambert Academic Publishing, Saarbrcken, 2010.

Abdus Saboor and Munir Ahmad, A bivariate gamma-type probability function using a confluent hypergeometric function of two variables, Pakistan Journal of Statistics, vol. 28, no. 1, 81–91, 2012.

C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, vol. 27, pp. 379–423, 623–656, 1948.

O. E. Smith and S. I. Adelfangt, Gust model based on the bivariate gamma probability distribution, Journal of Spacecraft and Rockets, vol. 18, no. 6, pp. 545–549, 1981.

O. E. Smith, S. I. Adelfang, and J. D. Tubbs, A bivariate gamma probability distribution with application to gust modeling, Space Science Laboratory, George C. Marshall Space Flight Center, NASA TM-82483, July 1982.

D. Vere-Jones, The infinite divisibility of a bivariate gamma distribution, Sankhya Ser. A, vol. 29, 421–422, 1967.

Yung Liang Tong, Probability inequalities in multivariate distributions, Probabilities and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980.

S.Yue, T. B. M. J. Ouarda, and B. Bobée, A review of bivariate gamma distribution for hydrological application, Journal of Hydrology, vol. 246, , no. 1-4, pp. 1–18, 2001.

L. Zhang and V. Singh, Copulas and their applications in water resources engineering, Cambridge: Cambridge University Press, 2019.

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.

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

Published
2020-08-25
How to Cite
Rafiei, M., Iranmanesh, A., & Nagar, D. k. (2020). A Bivariate Gamma Distribution Whose Marginals are Finite Mixtures of Gamma Distributions. Statistics, Optimization & Information Computing, 8(4), 950-971. https://doi.org/10.19139/soic-2310-5070-1001
Section
Research Articles