Bounds on Approximating Generalized Waring Distribution In A Generalized Waring Process

International Journal of Mathematics Trends and Technology (IJMTT)
© 2020 by IJMTT Journal
Volume-66 Issue-12
Year of Publication : 2020
Authors : Kanint Teerapabolarn


MLA Style: Kanint Teerapabolarn  "Bounds on Approximating Generalized Waring Distribution In A Generalized Waring Process" International Journal of Mathematics Trends and Technology 66.12 (2020):128-133. 

APA Style: Kanint Teerapabolarn(2020). Bounds on Approximating Generalized Waring Distribution In A Generalized Waring Process  International Journal of Mathematics Trends and Technology, 128-133.

This paper uses Stein's method for Poisson and negative binomial distributions to gether with the covariance associated with the generalized Warning random variable to determine error bounds for measuring the accuracy of approximations of generalized Waring distribution with parameters (a,kt,ρ) in a generalized Waring process, where a>0, k>0, ρ>0 and t≥0. The bounds in the present study are pointed out that(i) for c=ρ-1/k >0, the generalized Waring distribution can be approximated by the negative binomial distribution with parameters (a,c/c+t) when c and/or k are large and (ii) for λ=a/c>0, the generalized Waring distribution with parameters(a,kt,c) can be approximated by the Poisson distribution with mean λt when c is large or λt is small.


[1] A. D. Barbour, L. Holst and S. Janson, “Poisson Approximation” (Oxford Studies in Probability 2), Clarendon Press, Oxford, 1992.
[2] T. C. Brown, and M. J. Phillips, “Negative binomial approximation with Stein’s method”. Methodology and Computing in Applied Probability, vol. 1, 407-421, 1999.
[3] Q. L. Burrell, “The use of the generalized Waring process in modelling informatic data”. Scientometrics, vol. 64, pp. 247-270, 2005.
[4] T. Cacoullos, and V. Papathanasiou, “Characterization of distributions by variance bounds”. Statistics & Probability Letters, vol. 7, 351-356, 1989.
[5] L. H. Y. Chen, “Poisson approximation for dependent trials”. Annals of Probability, vol. 3, pp. 534-545, 1975.
[6] K. Jaioun, and K. Teerapabolarn, “A uniform bound on negative binomial approximation with w-functions”. Applied Mathematical Sciences, vol. 9, 2831-2841, 2015.
[7] B. Palumbo, “A generalization of some inequalities for the gamma function”. Journal of Computational and Applied Mathematics, vol. 8, 255-268, 1997.
[8] C. M. Stein, “A bound for the error in normal approximation to the distribution of a sum of dependent random variables”, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, California, Vol. 19-22, pp. 583-602, 1972.
[9] K. Teerapabolarn, and A. Boondirek, “Negative binomial approximation with Stein’s method and Stein’s identity”. International Mathematical Forum, vol. 5, 2541-2551, 2010.
[10] K. Teerapabolarn, “An improved bound for negative binomial approximation with z-functions”. AKCE International Journal of Graphs and Combinatorics, vol. 14, 287-294, 2017.
[11] W. S. Wang, “Some properties of k-gamma and k-beta functions”. ITM Web of Conferences, vol. 7, 1-6, 2016.
[12] E. Xekalaki, and M. Zografi, “The generalized Waring process and it application”. Communications in Statistics-Theory and Methods , vol. 37, 1835-1854, 2008.

Keywords : Generalized Waring process, Poisson approximation, Negative binomial approximation, Stein’s method.