Asymptotic properties of maximum likelihood estimates in the mixed Poisson model (Q1057011)

The asymptotic behavior of the maximum likelihood estimators (mle's) of the probabilities of a mixed Poisson distribution is considered, when a nonparametric mixing distribution is assumed. It is shown that if 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, then any finite set of mle's has the same limiting distribution as the corresponding set of sample proportions. Two \(\chi^ 2\)-type norms are used for studying the limiting behavior of the normalizing mle's and the sample proportions. Both estimators are shown to converge in probability in these norms to the vector of mixed probabilities at rate \(n^{-\epsilon}\) for any \(\epsilon >0\). Asymptotic normality of some linear combinations of the estimators are also established.