On the recovery of discrete probability densities from imperfect measurements

The problem of the estimation of a discrete probability density from independent observations is considered. For a wide class of noises, a method is given for estimating a probability density when the measurements are corrupted by additive noise. This method is shown to be consistent, and several bounds on the error are given. An application to the detection of a (nonparametric) random signal is discussed. Finally, the estimation of a probability density is considered where the measurements are noisy and some of the measurements are incorrect. This situation may arise when a machine collecting the data fails part of the time.