Estimating linear functionals in Poisson mixture models

This paper concerns the problem of estimating linear functionals of the mixing distribution from Poisson mixture observations. In particular, linear functionals for which a parametric rate of convergence cannot be achieved are studied. It appears that Gaussian functionals are rather easy to estimate. Estimation of the distribution functions is then considered by approximating this functional using Gaussian functionals. Finally, the case of smooth distribution functions is considered in order to deal with rather general linear functionals.