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When is the Conway–Maxwell–Poisson distribution infinitely divisible?

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  • Geng, Xi
  • Xia, Aihua

Abstract

An essential characteristic for a distribution to play a central role in limit theory is infinite divisibility. In this note, we prove that the Conway–Maxwell–Poisson (CMP) distribution is infinitely divisible iff it is the Poisson or geometric distribution. This explains that, despite its applications in a wide range of fields, there is no theoretical foundation for the CMP distribution to be a natural candidate for the law of small numbers.

Suggested Citation

  • Geng, Xi & Xia, Aihua, 2022. "When is the Conway–Maxwell–Poisson distribution infinitely divisible?," Statistics & Probability Letters, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:stapro:v:181:y:2022:i:c:s0167715221002261
    DOI: 10.1016/j.spl.2021.109264
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    References listed on IDEAS

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    4. Bo Li & Huiming Zhang & Jiao He, 2020. "Some characterizations and properties of COM-Poisson random variables," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(6), pages 1311-1329, March.
    5. Subrata Chakraborty & Tomoaki Imoto, 2016. "Extended Conway-Maxwell-Poisson distribution and its properties and applications," Journal of Statistical Distributions and Applications, Springer, vol. 3(1), pages 1-19, December.
    6. Tammy Harris & Joseph M. Hilbe & James W. Hardin, 2014. "Modeling count data with generalized distributions," Stata Journal, StataCorp LP, vol. 14(3), pages 562-579, September.
    7. Li Zhu & Kimberly F. Sellers & Darcy Steeg Morris & Galit Shmueli, 2017. "Bridging the Gap: A Generalized Stochastic Process for Count Data," The American Statistician, Taylor & Francis Journals, vol. 71(1), pages 71-80, January.
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