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A refined continuity correction for the negative binomial distribution and asymptotics of the median

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  • Frédéric Ouimet

    (California Institute of Technology
    Université de Montréal)

Abstract

In this paper, we prove a local limit theorem and a refined continuity correction for the negative binomial distribution. We present two applications of the results. First, we find the asymptotics of the median for a $$\textrm{NegativeBinomial}(r,p)$$ NegativeBinomial ( r , p ) random variable jittered by a $$\textrm{Uniform}(0,1)$$ Uniform ( 0 , 1 ) , which answers a problem left open in Coeurjolly and Trépanier (Metrika 83(7):837–851, 2020). This is used to construct a simple, robust and consistent estimator of the parameter p, when $$r > 0$$ r > 0 is known. The case where r is unknown is also briefly covered. Second, we find an upper bound on the Le Cam distance between negative binomial and normal experiments.

Suggested Citation

  • Frédéric Ouimet, 2023. "A refined continuity correction for the negative binomial distribution and asymptotics of the median," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(7), pages 827-849, October.
    • Handle: RePEc:spr:metrik:v:86:y:2023:i:7:d:10.1007_s00184-023-00897-2
    DOI: 10.1007/s00184-023-00897-2
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    References listed on IDEAS

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    1. Hamza, Kais, 1995. "The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions," Statistics & Probability Letters, Elsevier, vol. 23(1), pages 21-25, April.
    2. Pinelis, Iosif, 2021. "Monotonicity properties of the gamma family of distributions," Statistics & Probability Letters, Elsevier, vol. 171(C).
    3. R. Ven & N. Weber, 1993. "Bounds for the median of the negative binomial distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 40(1), pages 185-189, December.
    4. Mark Payton & Linda Young & J. Young, 1989. "Bounds for the difference between median and mean of beta and negative binomial distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 36(1), pages 347-354, December.
    5. Alzer, Horst, 2006. "A convexity property of the median of the gamma distribution," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1510-1513, August.
    6. Chen, Jeesen & Rubin, Herman, 1986. "Bounds for the difference between median and mean of gamma and poisson distributions," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 281-283, October.
    7. J. A. Adell & P. Jodrá, 2005. "The median of the poisson distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 61(3), pages 337-346, June.
    8. Jean-François Coeurjolly & Joëlle Rousseau Trépanier, 2020. "The median of a jittered Poisson distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(7), pages 837-851, October.
    9. Richard F Lyon, 2021. "On closed-form tight bounds and approximations for the median of a gamma distribution," PLOS ONE, Public Library of Science, vol. 16(5), pages 1-18, May.
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