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On Underdispersed Count Kernels for Smoothing Probability Mass Functions

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  • Célestin C. Kokonendji

    (Laboratoire de Mathématiques de Besançon UMR 6623 CNRS-UBFC, Université Bourgogne Franche-Comté, 16 Route de Gray, CEDEX, 25030 Besançon, France
    Laboratoire de Mathématiques et Connexes de Bangui, Université de Bangui, Av. des Martyrs, Bangui B.P. 908, Central African Republic
    These authors contributed equally to this work.)

  • Sobom M. Somé

    (Laboratoire d’Analyse Numérique Informatique et de BIOmathématique, Université Joseph KI-ZERBO, Ouagadougou 03 BP 7021, Burkina Faso
    Laboratoire Sciences et Techniques, Université Thomas SANKARA, Ouagadougou 12 BP 417, Burkina Faso
    These authors contributed equally to this work.)

  • Youssef Esstafa

    (Laboratoire Manceau de Mathématiques, Le Mans Université, Avenue Olivier Messiaen, CEDEX 09, 72085 Le Mans, France)

  • Marcelo Bourguignon

    (Departamento de Estatística, Universidade Federal do Rio Grande do Norte, Natal 59078-970, Brazil)

Abstract

Only a few count smoothers are available for the widespread use of discrete associated kernel estimators, and their constructions lack systematic approaches. This paper proposes the mean dispersion technique for building count kernels. It is only applicable to count distributions that exhibit the underdispersion property, which ensures the convergence of the corresponding estimators. In addition to the well-known binomial and recent CoM-Poisson kernels, we introduce two new ones such the double Poisson and gamma-count kernels. Despite the challenging problem of obtaining explicit expressions, these kernels effectively smooth densities. Their good performances are pointed out from both numerical and comparative analyses, particularly for small and moderate sample sizes. The optimal tuning parameter is here investigated by integrated squared errors. Also, the added advantage of faster computation times is really very interesting. Thus, the overall accuracy of two newly suggested kernels appears to be between the two old ones. Finally, an application including a tail probability estimation on a real count data and some concluding remarks are given.

Suggested Citation

  • Célestin C. Kokonendji & Sobom M. Somé & Youssef Esstafa & Marcelo Bourguignon, 2023. "On Underdispersed Count Kernels for Smoothing Probability Mass Functions," Stats, MDPI, vol. 6(4), pages 1-15, November.
    • Handle: RePEc:gam:jstats:v:6:y:2023:i:4:p:76-1240:d:1274193
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    References listed on IDEAS

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    1. Lynda Harfouche & Smail Adjabi & Nabil Zougab & Benedikt Funke, 2018. "Multiplicative bias correction for discrete kernels," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(2), pages 253-276, June.
    2. Walmes Marques Zeviani & Paulo Justiniano Ribeiro & Wagner Hugo Bonat & Silvia Emiko Shimakura & Joel Augusto Muniz, 2014. "The Gamma-count distribution in the analysis of experimental underdispersed data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(12), pages 2616-2626, December.
    3. Marcelo Bourguignon & Diego I. Gallardo & Rodrigo M. R. Medeiros, 2022. "A simple and useful regression model for underdispersed count data based on Bernoulli–Poisson convolution," Statistical Papers, Springer, vol. 63(3), pages 821-848, June.
    4. Hsiao, Cheng & Li, Qi & Racine, Jeffrey S., 2007. "A consistent model specification test with mixed discrete and continuous data," Journal of Econometrics, Elsevier, vol. 140(2), pages 802-826, October.
    5. Winkelmann, Rainer, 1995. "Duration Dependence and Dispersion in Count-Data Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(4), pages 467-474, October.
    6. Dexter Cahoy & Elvira Di Nardo & Federico Polito, 2021. "Flexible models for overdispersed and underdispersed count data," Statistical Papers, Springer, vol. 62(6), pages 2969-2990, December.
    7. Tammy Harris & Zhao Yang & James W. Hardin, 2012. "Modeling underdispersed count data with generalized Poisson regression," Stata Journal, StataCorp LLC, vol. 12(4), pages 736-747, December.
    8. Sobom M. Somé & Célestin C. Kokonendji & Nawel Belaid & Smail Adjabi & Rahma Abid, 2023. "Bayesian local bandwidths in a flexible semiparametric kernel estimation for multivariate count data with diagnostics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 32(3), pages 843-865, September.
    9. Marcelo Bourguignon & Rodrigo M. R. Medeiros, 2022. "A simple and useful regression model for fitting count data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 790-827, September.
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    1. Senga Kiessé, Tristan & Durrieu, Gilles, 2024. "On a discrete symmetric optimal associated kernel for estimating count data distributions," Statistics & Probability Letters, Elsevier, vol. 208(C).

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