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A simple and useful regression model for underdispersed count data based on Bernoulli–Poisson convolution

Author

Listed:
  • Marcelo Bourguignon

    (Universidade Federal do Rio Grande do Norte)

  • Diego I. Gallardo

    (Universidad de Atacama)

  • Rodrigo M. R. Medeiros

    (Universidade de São Paulo)

Abstract

Count data is often modeled using Poisson regression, although this probability model naturally restricts the conditional variance to be equal to the conditional mean (equidispersion property). While overdispersion has been intensively studied, there are few alternative models in the statistical literature for analyzing count data with underdispersion. The primary goal of this paper is to introduce a novel model based on Bernoulli-Poisson convolution for modelling count data that are underdispersed relative to the Poisson distribution. We study the statistical properties of the proposed model, and we provide a useful interpretation of the parameters. We consider a regression structure for both components based on a new parameterization indexed by mean and dispersion parameters. An expectation-maximization (EM) algorithm is proposed for parameter estimation and some diagnostic measures, based on the EM algorithm, are considered. Simulation studies are conducted to evaluate its finite sample performance. Finally, we illustrate the usefulness of the new regression model by an application.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:63:y:2022:i:3:d:10.1007_s00362-021-01253-0
    DOI: 10.1007/s00362-021-01253-0
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    References listed on IDEAS

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    1. Rigby, R.A. & Stasinopoulos, D.M. & Akantziliotou, C., 2008. "A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 381-393, December.
    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. Hong‐Tu Zhu & Sik‐Yum Lee, 2001. "Local influence for incomplete data models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(1), pages 111-126.
    4. 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.
    5. Marcelo Bourguignon & Diego I. Gallardo, 2020. "Reparameterized inverse gamma regression models with varying precision," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 74(4), pages 611-627, November.
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    Cited by:

    1. 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.

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