Product and Quotient of Independent Gauss Hypergeometric Variables
Main Article Content
Keywords
Appell's first hypergeometric function, beta distribution, Gauss hypergeometric distribution, quotient, transformation
Abstract
In this article, we have derived the probability density functions of the productand the quotient of two independent random variables having Gauss hypergeometricdistribution. These densities have been expressed in terms of Appell'srst hypergeometric function F1. Further, Renyi and Shannon entropies havealso been derived for the Gauss hypergeometric distribution.
MSC: 33Cxx, 33C65
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References
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[4] J. B. McDonald and Y. J. Xu. A generalization of the beta distribution with applications. Journal of Econometrics, ISSN: 0304-4076, 66(1&2), 133{152 (1995).
[5] D. K. Nagar and E. Zarrazola. Distributions of the product and the quotient of independent Kummer-beta variables. Scientiae Mathematicae Japonicae, ISSN: 1346{0862, EISSN: 1346{0447, 61(1), 109{117 (2005).
[6] C. Armero and M. Bayarri. Prior assessments for predictions in queues. The Statistician, ISSN: 1467{9884, 43(1), 139{153 (1994).
[7] Y. L. Luke. The special functions and their approximations, Vol. I. Mathematics in Science and Engineering, Vol. 53, ISBN: 0-124-59901-X, Academic Press, New York-London, 1969.
[8] Peter S. Fader and Bruce G. S. Hardie. A note on modelling underreported Poisson counts. Journal of Applied Statistics, PISSN: ISSN: 0266-4763, OISSN: 1360-0532, 27(8), 953{964 (2000).
[9] J.-Y. Dauxois. Bayesian inference for linear growth birth and death processes. Journal of Statistical Planning and Inference, ISSN: 0378-3758, 121(1), 1{19 (2004).
[10] Jos_e Mar__a Sarabia and Enrique Castillo. Bivariate distributions based on the generalized three-parameter beta distribution. Advances in distribution theory, order statistics, and inference, ISBN: 978-0-8176-4361-4, 85{110, Stat. Ind. Technol., Birkhauser Boston, Boston, MA, 2006.
[11] Liliam Carde~no, Daya K. Nagar and Luz Estela S_anchez. Beta type 3 distribution and its multivariate generalization. Tamsui Oxford Journal of Mathematical Sciences, ISSN: 1561-8307, 21(2), 225{241 (2005).
[12] Luz E. S_anchez and Daya K. Nagar. Distributions of the product and quotient of independent beta type 3 variables. Far East Journal of Theoretical Statistics, ISSN: 0972-0863, 17(2), 239{251 (2005).
[13] D. L. Libby and M. R. Novic. Multivariate generalized beta distributions with applications to utility assessment. Journal of Educational Statistics, ISSN: 0362-9791, 7(4), 271{294 (1982).
[14] J. J. Chen and M. R. Novick. Bayesian analysis for binomial models with generalized beta prior distributions. Journal of Educational Statistics, ISSN: 0362-9791, 9, 163{175 (1984).
[15] Gokarna Aryal and Saralees Nadarajah. Information matrix for beta distributions. Serdica Mathematical Journal, ISSN: 1310-6600, 30(4), 513{526 (2004).
[16] Saralees Nadarajah. Sums, products and ratios of generalized beta variables. Statistical Papers, PISSN: 0932-5026, EISSN: 1613-9798, 47, (1), 69{90 (2006).
[17] Daya K. Nagar and Erika Alejandra Rada-Mora. Properties of multivariate beta distributions. Far East Journal of Theoretical Statistics, ISSN: 0972{0863, 24(1), 73{94 (2008).
[18] T. Pham-Gia and Q. P. Duong. The generalized beta- and F-distributions in statistical modelling. Mathematics and Computer Modelling, ISSN: 0895-7177, 12(12), 1613{1625 (1989).
[19] H. M. Srivastava and P. W. Karlsson. Multiple Gaussian hypergeometric series, ISBN: 0-470-20100-2, Halsted Press [John Wiley & Sons], New York, 1985.
[20] C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal, ISSN : 0005-8580, 27, 379{423, 623{656 (1948).
[21] A. R_enyi. On measures of entropy and information. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability,
Volume 1: Contributions to the Theory of Statistics, ISSN: 0097- 0433, University of California Press, Berkeley, California, pp. 547{561 (1961).
[22] S. Nadarajah and K. Zografos. Expressions for R_enyi and Shannon entropies for bivariate distributions. Information Sciences, ISSN: 0020- 0255, 170(2-4), 173-{189 (2005).
[23] K. Zografos. On maximum entropy characterization of Pearson's type II and VII multivariate distributions. Journal of Multivariate Analysis, ISSN: 0047-259X, 71(1), 67-{75 (1999).
[24] K. Zografos and S. Nadarajah. Expressions for R_enyi and Shannon entropies for multivariate distributions. Statistics & Probability Letters, ISSN: 0167-7152, 71(1), 71-{84 (2005).