On Some Aspects of Generalized Extended Yule Distribution: Properties and Applications


Probability generating function
Model selection
Maximum likelihood estimation
Generalized likelihood ratio test

How to Cite

C. Satheesh Kumar, & S. Harisankar. (2020). On Some Aspects of Generalized Extended Yule Distribution: Properties and Applications. Journal of Advances in Applied &Amp; Computational Mathematics, 7, 49–56. https://doi.org/10.15377/2409-5761.2020.07.7


Martinez-Rodriguez (Comp. Statist. Dat. Anal., 2011) studied an extended version of the Yule distribution, namely “the extended Yule distribution (EYD)” which they obtained as a mixture of geometric distribution and generalized beta distribution. Through the present paper, we propose a generalized version of the EYD and named it “the generalized extended Yule distribution (GEYD)”. Several statistical properties of the distribution are obtained, including probability generating function (p.g.f), moments, recursion formulae etc. The maximum likelihood estimation of the parameters of the GEYD is discussed and fitted to two real-life data sets for illustrating its usefulness compared to the existing models. Further, the generalized likelihood ratio test procedure is considered for testing the significance of the parameters of the GEYD.



Yule GU. A mathematical theory of evolution based on the conclusion of Dr. J. C. Wills. Philosophical Society of Royal Society of London, Series B, 1920; 213: 21-87.

Simon HA. On a class of skew distribution. Biometrika, 1955; 42: 425-440. https://doi.org/10.1093/biomet/42.3-4.425

Kendall MG. Natural law in social sciences. Journal of Royal Statistical Society, Series A, 1961; 124: 1-28. https://doi.org/10.2307/2984139

Haight FA. Some statistical problems in connection with word association data. Journal of Mathematical Psychology 1966; 3(1): 217-233. https://doi.org/10.1016/0022-2496(66)90013-7

Xekalaki E. A property of Yule distribution and its applications. Communication in Statistics: Theory and Methods, 1983; 12: 140-150. https://doi.org/10.1080/03610928308828523

Jones JH. and Handcock MS. An assessment of preferential attachment as a mechanism for human sexual network formation. Proceedings of the Royal Society of London B: Biological Sciences, 2003; 270(1520): 1123-1128. https://doi.org/10.1098/rspb.2003.2369

Handcock MS. and Jones JH. Likelihood-based inference for stochastic models of sexual network formation. Theoretical Population Biology, 2004; 65: 413-422. https://doi.org/10.1016/j.tpb.2003.09.006

Dorogovtsev SN, Mendes JFF. and Samukhin AN. Structure of growing networks with preferential linking. Physical Review Letters, 2000; 85(21): 4633-4636. https://doi.org/10.1103/physrevlett.85.4633

Levene M, Fenner T, Loizou G. and Wheeldon R. A stochastic model for the evolution of the web. Computer Networks, 2002; 39(3): 277-287. https://doi.org/10.1016/s1389-1286(02)00209-8

Xekalaki E. and Panaretos J. On the association of the pareto and the yule distribution. Theory of Probability Its Applications, 1989; 33(1): 191-195. https://doi.org/10.1137/1133028

Singh H. and Vasudeva H. A characterization of exponential distribution by yule distribution. Journal of Indian Statistical Assosiation, 1984; 22: 93-96.

Prasad A. A new discrete distribution. Sankhya: The Indian Journal of Statistics, 1957; 17(4): 353-354.

Irwin JO. The place of mathematics in medical and biological statistic. Journal of Royal Statistical Society, 1968; 126: 1-41. https://doi.org/10.2307/2982445

Mishra A. On a generalized Yule distribution. Assam Statistical Review, 2009; 23: 140-150.

Kumar CS. and Harisankar S. On a modified Yule distribution. Statistica, 2018; 78(2): 169-180.

Kumar CS. and Harisankar S. On some aspects of a general class of Yule distribution and its applications. Communication in Statistics - Theory and Methods, 2019; 49(12): 2887-2897. https://doi.org/10.1080/03610926.2019.1584308

Martinez-Rodriguez AM, Saez-Castillo AJ. and CondeSanchez A. Modelling using an extended Yule distribution. Computational Statistics and Data Analysis, 2011; 55(1): 863-873. https://doi.org/10.1016/j.csda.2010.07.014

Mathai AM. and Haubold HJ. Special Functions for Applied Statistics. Springer, New York. 2008. https://doi.org/10.1007/978-0-387-75894-7

Johnson NL, Kemp AW. and Kotz S. Univariate Discrete Distributions (III). John Wiley Sons, New York. 2005; https://doi.org/10.1002/0471715816

Kumar CS. and Riyaz A. An extended zero-inflated logarithmic series distribution and its application. Journal of Applied Statistical Science, 2013; 21(1): 31-42.

Kumar CS. and Riyaz A. On zero-inflated logarithmic series distribution and its application. Statistica, 2013; 73(4): 477- 492. https://doi.org/10.1007/s10182-014-0229-1

Wagner GG, Burkhauser RV. and Behringer F. The English language public use file of the German socio-economic panel. Journal of Human Resource, 1993; 28: 429-433.

Bliss CI. and Fisher RA. Fitting the negative binomial distribution to biological data and note on the efficient fitting of the negative binomial. Biometrics, 1953; 9: 176-200. https://doi.org/10.2307/3001850

Rao CR. Minimum variance and the estimation of several parameters. In Mathematical Proceedings of the Cambridge Philosophical Society, 1947; 43: 280-283. Cambridge University Press. https://doi.org/10.1017/s0305004100023471

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