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Flexible models for overdispersed and underdispersed count data

Author

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  • Dexter Cahoy

    (University of Houston-Downtown)

  • Elvira Di Nardo

    (University of Torino)

  • Federico Polito

    (University of Torino)

Abstract

Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:62:y:2021:i:6:d:10.1007_s00362-021-01222-7
    DOI: 10.1007/s00362-021-01222-7
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    References listed on IDEAS

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    1. 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.
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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.
    2. Seng Huat Ong & Shin Zhu Sim & Shuangzhe Liu & Hari M. Srivastava, 2023. "A Family of Finite Mixture Distributions for Modelling Dispersion in Count Data," Stats, MDPI, vol. 6(3), pages 1-14, September.

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