A note on the mixed Poisson formulation of the Poisson-inverse Gaussian distribution

The Poisson-inverse Gaussian distribution is expressed in terms of a mixed Poisson distribution with the inverted gamma as mixing distribution. A simple approximation by Taylor expansion is derived for the Poisson-inverse Gaussian probabilities from this mixing expression. Based on this expansion, the asymptotic behaviour is considered. Shaban's approximation (Communications in Statistics, A, 1981, pp. 1389-1399) of the Poisson-inverse Gaussian probability by its limiting case is examined, and it is shown that this approximation can be improved by using the inverted gamma mixing formulation.