Filomat 2016 Volume 30, Issue 4, Pages: 953-960
https://doi.org/10.2298/FIL1604953S
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Approximating the Conway-Maxwell-Poisson normalizing constant
Şimşek Burçin (University of Pittsburgh, Department of Statistics, USA)
Iyengar Satish (University of Pittsburgh, Department of Statistics, USA)
The Conway-Maxwell-Poisson is a two-parameter family of distributions on the
nonnegative integers. Its parameters λ and ν model the intensity and the
dispersion, respectively. Its normalizing constant is not always easy to
compute, so good approximations are needed along with an assessment of their
error. Shmueli, et al. [11] derived an approximation assuming that ν is an
integer, and gave an estimate of the relative error. Their numerical work
showed that their approximation performs well in some parameter ranges but
not in others. Our aims are to show that this approximation applies to all
real ν > 0; to provide correction terms to this approximation; and to give
different approximations for ν very small and very large. We then investigate
the error terms numerically to assess our approximations. In parameter ranges
for which Shmueli’s approximation does poorly we show that our correction
terms or alternative approximations give considerable improvement.
Keywords: asymptotic expansion, Gaussian approximation, generalized hypergeometric function, geometric distribution, modified Bessel function, relative error