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An asymptotic expansion for the normalizing constant of the Conway–Maxwell–Poisson distribution

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  • Robert E. Gaunt

    (The University of Manchester)

  • Satish Iyengar

    (University of Pittsburgh)

  • Adri B. Olde Daalhuis

    (The University of Edinburgh)

  • Burcin Simsek

    (University of Pittsburgh)

Abstract

The Conway–Maxwell–Poisson distribution is a two-parameter generalization of the Poisson distribution that can be used to model data that are under- or over-dispersed relative to the Poisson distribution. The normalizing constant $$Z(\lambda ,\nu )$$ Z ( λ , ν ) is given by an infinite series that in general has no closed form, although several papers have derived approximations for this sum. In this work, we start by using probabilistic argument to obtain the leading term in the asymptotic expansion of $$Z(\lambda ,\nu )$$ Z ( λ , ν ) in the limit $$\lambda \rightarrow \infty $$ λ → ∞ that holds for all $$\nu >0$$ ν > 0 . We then use an integral representation to obtain the entire asymptotic series and give explicit formulas for the first eight coefficients. We apply this asymptotic series to obtain approximations for the mean, variance, cumulants, skewness, excess kurtosis and raw moments of CMP random variables. Numerical results confirm that these correction terms yield more accurate estimates than those obtained using just the leading-order term.

Suggested Citation

  • Robert E. Gaunt & Satish Iyengar & Adri B. Olde Daalhuis & Burcin Simsek, 2019. "An asymptotic expansion for the normalizing constant of the Conway–Maxwell–Poisson distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 163-180, February.
  • Handle: RePEc:spr:aistmt:v:71:y:2019:i:1:d:10.1007_s10463-017-0629-6
    DOI: 10.1007/s10463-017-0629-6
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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.
    2. Saralees Nadarajah, 2009. "Useful moment and CDF formulations for the COM–Poisson distribution," Statistical Papers, Springer, vol. 50(3), pages 617-622, June.
    3. Boatwright, Peter & Borle, Sharad & Kadane, Joseph B., 2003. "A Model of the Joint Distribution of Purchase Quantity and Timing," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 564-572, January.
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    2. Morris, Darcy Steeg & Raim, Andrew M. & Sellers, Kimberly F., 2020. "A Conway–Maxwell-multinomial distribution for flexible modeling of clustered categorical data," Journal of Multivariate Analysis, Elsevier, vol. 179(C).

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