Smooth Versus Likelihood Estimation for a Class of Mixtures of Discrete Distributions

The problem of estimating the mixing distribution Q of a power series distributions (PSD’s) mixture model is considered. In the first part of the chapter, some results on existence, support size, uniqueness and convergence of the (asymptotic) maximum likelihood estimator are presented. It is allowed for the mixture model to be misspecified. In the second part of the chapter, we look at the nonparametric Bayesian approach for estimating mixtures. In this approach, a Dirichlet random probability can be considered as a prior on the space of the mixing distributions. The form of the posterior distribution of the Dirichlet process is known to be very complicated, and by consequence, the Bayesian estimators are not easily computable. However, it has been remarked before that, conditioning on some additional information, the form of the nonparametric Bayesian estimator of the mixing distribution is much simpler. In a PSD’s mixture model, this remark gives an idea for a smooth estimator of the mixing distribution which has the form of a finite mixture of natural conjugate priors of the PSD’s compound family. This smooth estimator is, in a sense, a kind of kernel estimator. Convergence results are obtained for this estimator. It turns out that, under quite mild conditions, the proposed estimator is an asymptotic maximum likelihood estimator. Some simulations show the behavior of this estimator for small and moderate sample sizes in the case of Poisson mixtures.