Asymptotic Properties of Maximum Likelihood Estimates in the Mixed Poisson Model

Abstract

This paper considers the asymptotic behavior of the maximum likelihood estimators (mle's) of the probabilities of a mixed Poisson distribution with a nonparametric mixing distribution. The vector of estimated probabilities is shown to converge in probability to the vector of mixed probabilities at rate $n^{1/2-\varepsilon}$ for any $\varepsilon > 0$ under a generalized $\chi^2$ distance function. It is then shown that any finite set of the mle's has the same joint limiting distribution as does the corresponding set of sample proportions when the support of the mixing distribution $G_0$ is an infinite set with a known upper bound and $G_0$ satisfies a certain condition at zero.