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

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## 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 \(Z(\lambda ,\nu )\) 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 \(Z(\lambda ,\nu )\) in the limit \(\lambda \rightarrow \infty \) that holds for all \(\nu >0\). 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.

## Keywords

Conway–Maxwell–Poisson distribution Normalizing constant Approximation Asymptotic series Generalized hypergeometric function Stein’s method## Notes

### Acknowledgements

RG is supported by a Dame Kathleen Ollerenshaw Research Fellowship. SI is supported by a grant from the National Institute of Mental Health (5R01 MH060952-09). AOD is supported by a research grant (GRANT 11863412/70NANB15H221) from the National Institute of Standards and Technology. The authors would like to thank the referees for their helpful comments and suggestions.

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