A Characterization of Convoluted Poisson Distributions with Applications to Estimation

Define X as the sum of a Poisson random variable Y with parameter θ and a nonnegative integer valued variable Z independent of Y. If the distribution of Z is known, X has a one-parameter convoluted Poisson distribution. Such distributions are here characterized as solutions to a system of differential equations specifying the rate of change of the cumulative distribution function relative to a parametric change. The characterization is used in point and interval estimation of θ. It is shown that for a certain subclass of convoluted Poisson distributions, the maximum likelihood estimate of θ is the unique solution of the likelihood equation and is easily approximated numerically.