  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.
