Consistent Density Estimation Under Discrete Mixture Models

This work considers a problem of estimating a mixing probability density f in the setting of discrete mixture models. The paper consists of three parts. The first part focuses on the construction of an L1 consistent estimator of f. In particular, under the assumptions that the probability measure μ of the observation is atomic, and the map from f to μ is bijective, it is shown that there exists an estimator fn such that for every density f limn→ ∞ mathbbE left[ int |fn -f | right]=0. The second part discusses the implementation details. Specifically, it is shown that the consistency for every f can be attained with a computationally feasible estimator. The third part, as a study case, considers a Poisson mixture model. In particular, it is shown that in the Poisson noise setting, the bijection condition holds and, hence, estimation can be performed consistently for every f.