# 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.

## References

- Boatwright, P., Borle, S., Kadane, J. B. (2003). A model of the joint distribution of purchase quantity and timing.
*Journal of the American Statistical Association*,*98*, 564–572.Google Scholar - Conway, R. W., Maxwell, W. L. (1962). A queueing model with state dependent service rate.
*Journal of Industrial Engineering*,*12*, 132–136.Google Scholar - Daly, F., Gaunt, R. E. (2016). The Conway–Maxwell–Poisson distribution: Distributional theory and approximation.
*ALEA*:*Latin American Journal of Probability and Mathematical Statistics*,*13*, 635–658.Google Scholar - Gaunt, R. E. (2015). Stein’s method for functions of multivariate normal random variables. arXiv:1507.08688.
- Gillispie, S. B., Green, C. G. (2015). Approximating the Conway–Maxwell–Poisson distribution normalizing constant.
*Statistics*,*49*, 1062–1073.Google Scholar - Hazelwinkel, M. (1997).
*Encyclopedia of mathematics, Supplement I*. Dordrehct: Kluwer Academic Publishers.Google Scholar - Hinch, E. J. (1991).
*Perturbation methods*. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar - Kadane, J. B., Shmueli, G., Minka, T. P., Borle, S., Boatwright, P. (2006). Conjugate analysis of the Conway–Maxwell–Poisson distribution.
*Bayesian Analysis*,*1*, 403–420.Google Scholar - Lin, Y., Wong, R. (2017). Asymptotics of generalized hypergeometric functions.
*Frontiers of orthogonal polynomials and q-series*. Singapore: World Scientific. (**preprint**).Google Scholar - Nadarajah, S. (2009). Useful moment and CDF formulations for the COM-Poisson distribution.
*Statistical Papers*,*50*, 617–622.MathSciNetCrossRefzbMATHGoogle Scholar - NIST Digital Library of Mathematical Functions. (2016). Olver, F. W. J., Olde Daalhuis, A. B., Lozier, D. W., Schneider, B. I., Boisvert, R. F., Clark, C. W., Miller, B. R., Saunders, B. V. (eds.). http://dlmf.nist.gov/. Release 1.0.12 of 2016-09-09.
- Olver, F. W. J. (1974).
*Asymptotics and special functions*. New York: Academic Press.zbMATHGoogle Scholar - Paris, R. B., Kaminski, D. (2001).
*Asymptotics and Mellin–Barnes integrals*. Cambridge: Cambridge University Press.Google Scholar - Pogány, T. K. (2016). Integral form of the COM-Poisson renormalization constant.
*Statistics and Probability Letters*,*116*, 144–145.MathSciNetCrossRefzbMATHGoogle Scholar - Rodrigues, J., de Castro, M., Cancho, V. G., Balakrishnan, N. (2009). COM-Poisson cure rate survival models and an application to a cutaneous melanoma data.
*Journal of Statistical Planning and Inference*,*139*, 3605–3611.Google Scholar - Sellers, K. F., Shmueli, G. (2010). A flexible regression model for count data.
*Annals of Applied Statistics*,*4*, 943–961.Google Scholar - Sellers, K. F., Borle, S., Shmueli, G. (2012). The COM-Poisson model for count data: A survey of methods and applications.
*Applied Stochastic Models in Business and Industry*,*28*, 104–116.Google Scholar - Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., Boatwright, P. (2005). A useful distribution for fitting discrete data: Revival of the COM-Poisson.
*Journal of the Royal Statistical Society*:*Series C*,*54*, 127–142.Google Scholar - Şimşek, B., Iyengar, S. (2016). Approximating the Conway–Maxwell–Poisson normalizing constant.
*Filomat*,*30*, 953–960.Google Scholar - Stein, C. (1986).
*Approximate computation of expectations*. Hayward, California: IMS.Google Scholar - Weniger, E. J. (2010). Summation of divergent power series by means of factorial series.
*Applied Numerical Mathematics*,*60*, 1429–1441.MathSciNetCrossRefzbMATHGoogle Scholar