Skip to main content
Log in

Poisson Approximation for a Sum of Negative Binomial Random Variables

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barbour, A.D., Holst, L., Janson, S.: Poisson Approximation, Oxford Studies in Probability 2. Clarendon Press, Oxford (1992)

    MATH  Google Scholar 

  2. Chen, L.H.Y.: Poisson approximation for dependent trials. Ann. Probab. 3, 534–545 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Stein, C.M.: A bound for the error in normal approximation to the distribution of a sum of dependent random variables. In: Proceedings Sixth Berkeley Symposium Mathematical Statistics and Probability 3, 583–602 (1972)

  4. Teerapabolarn, K.: An improvement of Poisson approximation for sums of dependent Bernoulli random variables. Commun. Stat. Theory Methods 43, 1758–1777 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Vellaisamy, P., Upadhye, N.S.: Compound negative binomial approximations for sums of random variables. Probab. Math. Stat. 29, 205–226 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is very grateful to the referees for valuable comments which have led to the improvement in the presentation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. Teerapabolarn.

Additional information

Communicated by Anton Abdulbasah Kamil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-016-0328-0

Keywords

Mathematics Subject Classification

Navigation