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Confidence Intervals for Negative Binomial Random Variables of High Dispersion

Author

Listed:
  • Shilane David

    (Stanford University)

  • Evans Steven N

    (University of California, Berkeley)

  • Hubbard Alan E.

    (University of California, Berkeley)

Abstract

We consider the problem of constructing confidence intervals for the mean of a Negative Binomial random variable based upon sampled data. When the sample size is large, it is a common practice to rely upon a Normal distribution approximation to construct these intervals. However, we demonstrate that the sample mean of highly dispersed Negative Binomials exhibits a slow convergence in distribution to the Normal as a function of the sample size. As a result, standard techniques (such as the Normal approximation and bootstrap) will construct confidence intervals for the mean that are typically too narrow and significantly undercover at small sample sizes or high dispersions. To address this problem, we propose techniques based upon Bernstein's inequality or the Gamma and Chi Square distributions as alternatives to the standard methods. We investigate the impact of imposing a heuristic assumption of boundedness on the data as a means of improving the Bernstein method. Furthermore, we propose a ratio statistic relating the Negative Binomial's parameters that can be used to ascertain the applicability of the Chi Square method and to provide guidelines on evaluating the length of all proposed methods. We compare the proposed methods to the standard techniques in a variety of simulation experiments and consider data arising in the serial analysis of gene expression and traffic flow in a communications network.

Suggested Citation

  • Shilane David & Evans Steven N & Hubbard Alan E., 2010. "Confidence Intervals for Negative Binomial Random Variables of High Dispersion," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-41, March.
  • Handle: RePEc:bpj:ijbist:v:6:y:2010:i:1:n:10
    DOI: 10.2202/1557-4679.1164
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    References listed on IDEAS

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    1. J. O. Lloyd-Smith & S. J. Schreiber & P. E. Kopp & W. M. Getz, 2005. "Superspreading and the effect of individual variation on disease emergence," Nature, Nature, vol. 438(7066), pages 355-359, November.
    2. Ferreri, Carlo, 1997. "On the ML-estimator of the positive and negative two-parameter binomial distribution," Statistics & Probability Letters, Elsevier, vol. 33(2), pages 129-134, April.
    3. Aragón, Jorge & Eberly, David & Eberly, Shelly, 1992. "Existence and uniqueness of the maximum likelihood estimator for the two-parameter negative binomial distribution," Statistics & Probability Letters, Elsevier, vol. 15(5), pages 375-379, December.
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    Cited by:

    1. Bao-Anh Dang & K. Krishnamoorthy, 2022. "Confidence intervals, prediction intervals and tolerance intervals for negative binomial distributions," Statistical Papers, Springer, vol. 63(3), pages 795-820, June.
    2. Ranjani Atukorala & Maxwell L. King & Sivagowry Sriananthakumar, 2014. "Applications of Information Measures to Assess Convergence in the Central Limit Theorem," Monash Econometrics and Business Statistics Working Papers 29/14, Monash University, Department of Econometrics and Business Statistics.

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