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Abstract
The Conway–Maxwell–Poisson distribution is a twoparameter generalization of the Poisson distribution that can be used to model data that are under or overdispersed relative to the Poisson distribution. The normalizing constant 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(λ, ν) in the limit λ → ∞ that holds for all 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 leadingorder term.
Original language  English 

Pages (fromto)  163180 
Number of pages  18 
Journal  Annals of the Institute of Statistical Mathematics 
Volume  71 
Issue number  1 
Early online date  15 Nov 2017 
DOIs  
Publication status  Published  Feb 2019 
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 1 Finished

Rigorous and Presentable Asymptotics for Special Functions and Orthogonal Polynomials
1/09/15 → 31/08/20
Project: Research