IDEAS home Printed from https://ideas.repec.org/a/spr/metrik/v86y2023i7d10.1007_s00184-023-00897-2.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00184-023-00897-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00184-023-00897-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. V. J. García & M. Martel & F.J. Vázquez-Polo, 2015. "Complementary information for skewness measures," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(4), pages 442-459, November.
    3. Phillips, Peter C.B. & Gao, Wayne Yuan, 2017. "Structural inference from reduced forms with many instruments," Journal of Econometrics, Elsevier, vol. 199(2), pages 96-116.
    4. Ding, S. & Koole, G. & van der Mei, R.D., 2015. "On the estimation of the true demand in call centers with redials and reconnects," European Journal of Operational Research, Elsevier, vol. 246(1), pages 250-262.
    5. 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.
    6. Hossein Abouee‐Mehrizi & Oded Berman & Hassan Shavandi & Ata G. Zare, 2011. "An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(8), pages 713-730, December.
    7. Deligiannis, Michalis & Liberopoulos, George & Benioudakis, Myron, 2023. "Dynamic supplier competition and cooperation for buyer loyalty on service," International Journal of Production Economics, Elsevier, vol. 255(C).
    8. Jean-François Coeurjolly, 2017. "Median-based estimation of the intensity of a spatial point process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 303-331, April.
    9. Alexander Katzur & Udo Kamps, 2020. "Classification using sequential order statistics," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(1), pages 201-230, March.
    10. Hofer, Vera & Leitner, Johannes, 2017. "Relative pricing of binary options in live soccer betting markets," Journal of Economic Dynamics and Control, Elsevier, vol. 76(C), pages 66-85.
    11. Alzer, Horst, 2006. "A convexity property of the median of the gamma distribution," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1510-1513, August.
    12. Adam Kasperski & Paweł Zieliński, 2009. "A randomized algorithm for the min-max selecting items problem with uncertain weights," Annals of Operations Research, Springer, vol. 172(1), pages 221-230, November.
    13. Luis Mendo, 2012. "Estimation of a probability in inverse binomial sampling under normalized linear-linear and inverse-linear loss," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 656-675, December.
    14. Noorizadegan, Mahdi & Chen, Bo, 2018. "Vehicle routing with probabilistic capacity constraints," European Journal of Operational Research, Elsevier, vol. 270(2), pages 544-555.
    15. Adell, José A. & Jodrá, Pedro, 2005. "Sharp estimates for the median of the [Gamma](n+1,1) distribution," Statistics & Probability Letters, Elsevier, vol. 71(2), pages 185-191, February.
    16. Dmitriev, Daniil & Zhukovskii, Maksim, 2021. "On monotonicity of Ramanujan function for binomial random variables," Statistics & Probability Letters, Elsevier, vol. 177(C).
    17. Pinelis, Iosif, 2021. "Monotonicity properties of the gamma family of distributions," Statistics & Probability Letters, Elsevier, vol. 171(C).
    18. 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.
    19. Noguchi, Kimihiro & Ward, Mayla C., 2024. "Asymptotic optimality of the square-root transformation on the gamma distribution using the Kullback–Leibler information number criterion," Statistics & Probability Letters, Elsevier, vol. 210(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metrik:v:86:y:2023:i:7:d:10.1007_s00184-023-00897-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.