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Modeling Under-Dispersed Count Data by the Generalized Poisson Distribution via Two New MM Algorithms

Author

Listed:
  • Xun-Jian Li

    (Department of Statistics and Data Science, Southern University of Science and Technology, Shenzhen 518055, China
    These authors contributed equally to this work.)

  • Guo-Liang Tian

    (Department of Statistics and Data Science, Southern University of Science and Technology, Shenzhen 518055, China
    These authors contributed equally to this work.)

  • Mingqian Zhang

    (Department of Statistics and Data Science, Southern University of Science and Technology, Shenzhen 518055, China)

  • George To Sum Ho

    (Department of Supply Chain and Information Management, The Hang Seng University of Hong Kong, Shatin, N.T., Hong Kong, China)

  • Shuang Li

    (Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China)

Abstract

Under-dispersed count data often appear in clinical trials, medical studies, demography, actuarial science, ecology, biology, industry and engineering. Although the generalized Poisson (GP) distribution possesses the twin properties of under- and over-dispersion, in the past 50 years, many authors only treat the GP distribution as an alternative to the negative binomial distribution for modeling over-dispersed count data. To our best knowledge, the issues of calculating maximum likelihood estimates (MLEs) of parameters in GP model without covariates and with covariates for the case of under-dispersion were not solved up to now. In this paper, we first develop a new minimization–maximization (MM) algorithm to calculate the MLEs of parameters in the GP distribution with under-dispersion, and then we develop another new MM algorithm to compute the MLEs of the vector of regression coefficients for the GP mean regression model for the case of under-dispersion. Three hypothesis tests (i.e., the likelihood ratio, Wald and score tests) are provided. Some simulations are conducted. The Bangladesh demographic and health surveys dataset is analyzed to illustrate the proposed methods and comparisons with the existing Conway–Maxwell–Poisson regression model are also presented.

Suggested Citation

  • Xun-Jian Li & Guo-Liang Tian & Mingqian Zhang & George To Sum Ho & Shuang Li, 2023. "Modeling Under-Dispersed Count Data by the Generalized Poisson Distribution via Two New MM Algorithms," Mathematics, MDPI, vol. 11(6), pages 1-24, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1478-:d:1100540
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    References listed on IDEAS

    as
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    3. Kimberly F. Sellers & Darcy S. Morris, 2017. "Underdispersion models: Models that are “under the radar”," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(24), pages 12075-12086, December.
    4. Seth D. Guikema & Jeremy P. Goffelt, 2008. "A Flexible Count Data Regression Model for Risk Analysis," Risk Analysis, John Wiley & Sons, vol. 28(1), pages 213-223, February.
    5. Galit Shmueli & Thomas P. Minka & Joseph B. Kadane & Sharad Borle & Peter Boatwright, 2005. "A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(1), pages 127-142, January.
    6. Angers, Jean-Francois & Biswas, Atanu, 2003. "A Bayesian analysis of zero-inflated generalized Poisson model," Computational Statistics & Data Analysis, Elsevier, vol. 42(1-2), pages 37-46, February.
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