Nonparametric estimation of mixing densities for discrete distributions

By a mixture density is meant a density of the form $\pi_{\mu}(\cdot)=\int\pi_{\theta}(\cdot)\times\mu(d\theta)$, where $(\pi_{\theta})_{\theta\in\Theta}$ is a family of probability densities and $\mu$ is a probability measure on $\Theta$. We consider the problem of identifying the unknown part of this model, the mixing distribution $\mu$, from a finite sample of independent observations from $\pi_{\mu}$. Assuming that the mixing distribution has a density function, we wish to estimate this density within appropriate function classes. A general approach is proposed and its scope of application is investigated in the case of discrete distributions. Mixtures of power series distributions are more specifically studied. Standard methods for density estimation, such as kernel estimators, are available in this context, and it has been shown that these methods are rate optimal or almost rate optimal in balls of various smoothness spaces. For instance, these results apply to mixtures of the Poisson distribution parameterized by its mean. Estimators based on orthogonal polynomial sequences have also been proposed and shown to achieve similar rates. The general approach of this paper extends and simplifies such results. For instance, it allows us to prove asymptotic minimax efficiency over certain smoothness classes of the above-mentioned polynomial estimator in the Poisson case. We also study discrete location mixtures, or discrete deconvolution, and mixtures of discrete uniform distributions.