Abstract
The Stein–Chen method is used to determine uniform and non-uniform bounds on the ratio between the distribution function of a sum of independent negative binomial random variables and a Poisson distribution function with mean \(\lambda = \sum _{i=1}^nr_iq_i\), where \(r_i\) and \(p_i=1-q_i\) are parameters of each negative binomial distribution. With these bounds, it indicates that the Poisson distribution function with this mean can be used as an estimate of the independent summands when all \(q_i\) are small or \(\lambda \) is small. Finally, some numerical examples for each result are given.
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The author is very grateful to the referees for valuable comments which have led to the improvement in the presentation.
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Communicated by Anton Abdulbasah Kamil.
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Teerapabolarn, K. Poisson Approximation for a Sum of Negative Binomial Random Variables. Bull. Malays. Math. Sci. Soc. 40, 931–939 (2017). https://doi.org/10.1007/s40840-016-0328-0
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DOI: https://doi.org/10.1007/s40840-016-0328-0