Binomial Approximation with Stein’s Method and w–Functions
Abstract
This study uses Stein’s method and -functions to determine a non-uniform bound for approximating the cumulative distribution function of a non-negative integer-valued random variable by the binomial cumulative distribution function that is in the form of the distance between the both cumulative distribution functions. The obtained non-uniform bound can be used as a new alternative criterion for measuring the accuracy of the binomial approximation. For theoretical applications, this study uses the result to approximate the cumulative distribution functions of hypergeometric, Polya and negative hypergeometric random variables. Keywords : binomial approximation, non-uniform bound, Stein’s method, functions .References
Barbour, A. D., Holst, L., & Janson, S. (1992). Poisson approximation (Oxford Studies in Probability 2). Oxford: Clarendon Press.
Cacoullos, T., & Papathanasiou, V. (1989). Characterization of distributions by variance bounds. Statistics &
Probability Letters, 7(5), 351-356.
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Annals of Probability, 3(3), 535-545.
Ehm, W., (1991). Binomial approximation to the Poisson binomial distribution. Statistics & Probability Letters, 11(1), 7-16.
Teerapabolarn, K., & Sae-Jeng, K. (2017). A non-uniform bound on binomial approximation to the beta binomial cumulative distribution function. Retrieved January 3, 2018, from http://rdo.psu.ac.th/sjstweb/Ar-Press/ 60-Nov/23.pdf.
Majsnerowska, M. (1998). A note on Poisson approximation by w-functions. Applicationes Mathematicae, 25(3), 387-392.
Prukpousana, K., & Teerapabolarn, K. (2010). Binomial approximation to the negative hypergeometric distribution. KKU Science Journal, 38(4), 606-616. (in Thai)
Soon, S. Y. T. (1996). Binomial approximation for dependent indicators. Statistica Sinica, 6(3), 703-714.
Stein, C. M. (1972). A bound for the error in normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (pp. 583-602). Berkeley: University of California Press.
Stein, C. M. (1986). Approximate computation of expectations. Hayward California: IMS.
Teerapabolarn, K. (2011). A non-uniform bound on pointwise approximation of generalized binomial distribution by binomial diatribution. Srinakharinwirot Science Journal, 27(1), 37-52. (in Thai)
Teerapabolarn, K., & Wongkasem, P. (2011). On pointwise binomial approximation by w-functions. International Journal of Pure and Applied Mathematics, 71(1), 57-66.
Wongkasem, P., Teerapabolarn, K., & Gulasirima, R. (2008). On approximating a generalized binomial by
Cacoullos, T., & Papathanasiou, V. (1989). Characterization of distributions by variance bounds. Statistics &
Probability Letters, 7(5), 351-356.
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Annals of Probability, 3(3), 535-545.
Ehm, W., (1991). Binomial approximation to the Poisson binomial distribution. Statistics & Probability Letters, 11(1), 7-16.
Teerapabolarn, K., & Sae-Jeng, K. (2017). A non-uniform bound on binomial approximation to the beta binomial cumulative distribution function. Retrieved January 3, 2018, from http://rdo.psu.ac.th/sjstweb/Ar-Press/ 60-Nov/23.pdf.
Majsnerowska, M. (1998). A note on Poisson approximation by w-functions. Applicationes Mathematicae, 25(3), 387-392.
Prukpousana, K., & Teerapabolarn, K. (2010). Binomial approximation to the negative hypergeometric distribution. KKU Science Journal, 38(4), 606-616. (in Thai)
Soon, S. Y. T. (1996). Binomial approximation for dependent indicators. Statistica Sinica, 6(3), 703-714.
Stein, C. M. (1972). A bound for the error in normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (pp. 583-602). Berkeley: University of California Press.
Stein, C. M. (1986). Approximate computation of expectations. Hayward California: IMS.
Teerapabolarn, K. (2011). A non-uniform bound on pointwise approximation of generalized binomial distribution by binomial diatribution. Srinakharinwirot Science Journal, 27(1), 37-52. (in Thai)
Teerapabolarn, K., & Wongkasem, P. (2011). On pointwise binomial approximation by w-functions. International Journal of Pure and Applied Mathematics, 71(1), 57-66.
Wongkasem, P., Teerapabolarn, K., & Gulasirima, R. (2008). On approximating a generalized binomial by
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Published
2018-02-23
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