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Negative Binomial Approximation with Stein's Method

Author

Listed:
  • Timothy C. Brown

    (University of Melbourne)

  • M. J. Phillips

    (University of Melbourne)

Abstract

Bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws. For an arbitrary discrete random variable written as a sum of indicators, an upper bound of coupling form is expressed as an average of terms each of which measures the difference between the effect of particular indicator being one and the value of a geometrically distributed random variable. When a monotone coupling exists a lower bound can also be shown. Application of these results is illustrated with the example of the Po´lya distribution for which the rate of approach to the negative binomial limit is found.

Suggested Citation

  • Timothy C. Brown & M. J. Phillips, 1999. "Negative Binomial Approximation with Stein's Method," Methodology and Computing in Applied Probability, Springer, vol. 1(4), pages 407-421, December.
    • Handle: RePEc:spr:metcap:v:1:y:1999:i:4:d:10.1023_a:1010094221471
    DOI: 10.1023/A:1010094221471
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    Cited by:

    1. Kumar, Amit N. & Kumar, Poleen, 2024. "A negative binomial approximation to the distribution of the sum of maxima of indicator random variables," Statistics & Probability Letters, Elsevier, vol. 208(C).
    2. Aihua Xia & Fuxi Zhang, 2009. "Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance," Journal of Theoretical Probability, Springer, vol. 22(2), pages 294-310, June.

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