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International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 65 | Issue 11 | Year 2019 | Article Id. IJMTT-V65I11P510 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I11P510

A Non-uniform Bound on Poisson Approximation for Random Sums of Negative Binomial Random Variables


Kanint Teerapabolarn
Abstract

This paper uses the Stein-Chen method to determine a non-uniform bound for the point metric between the distribution of random sums of independent negative binomial random variables and a Poisson distribution. Three examples are provided to illustrate applications of the result obtained.

Keywords
Negative binomial distribution, Poisson approximation, point metric, random sums, Stein-Chen method.
References

[1] A. D. Barbour, L. Holst and S. Janson, S. “Poisson Approximation” (Oxford Studies in Probability 2), Clarendon Press, Oxford, 1992.
[2] L. H. Y. Chen, “Poisson approximation for dependent trials”. Annals of Probability, vol. 3, pp. 534-545, 1975.
[3] S. Kongudomthrap and N. Chaidee, N. 2012. “Bounds in Poisson approximation of random sums of Bernoulli random variables”, Journal of Mathematics Research, vol. 4, pp. 29-35. 2012.
[4] R. Kun and K. Teerapabolarn, “A pointwise Poisson approximation by w-functions”, Applied Mathematical Sciences, vol. 6, pp. 5029-5037, 2012.
[5] 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.
[6] K. Teerapabolarn, “A new bound on Poisson approximation for random sums of Bernoulli random variables”, International Journal of Pure and Applied Mathematics, vol. 89, pp. 141-146, 2013.
[7] K. Teerapabolarn, “A pointwise Poisson approximation for random sums of geometric random variables”, International Journal of Pure and Applied Mathematics, vol. 89, pp. 353-356, 2013.
[8] K. Teerapabolarn, “A non-uniform bound on Poisson approximation for random sums of geometric random variables”, International Journal of Pure and Applied Mathematics, vol. 90, pp. 5-9, 2014.
[9] K. Teerapabolarn, “Poisson approximation for random sums of independent negative binomial random variables”, International Journal of Pure and Applied Mathematics, vol. 93, pp. 783-787, 2014.
[10] K. Teerapabolarn, “A non-uniform bound on Poisson approximation for a sum of negative binomial random variables”, Songklanakarin Journal of Science and Technology, vol. 3, pp. 355-358, 2017.
[11] N. Yannaros, “Poisson approximation for random sums of Bernoulli random variables”, Statistics & Probability Letters, vol. 11, pp. 161-165, 1991.

Citation :

Kanint Teerapabolarn, "A Non-uniform Bound on Poisson Approximation for Random Sums of Negative Binomial Random Variables," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 11, pp. 99-102, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I11P510

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