Consistent Density Estimation Under Discrete Mixture Models

Abstract

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 L_1 consistent estimator of f. In particular, under the assumptions that the probability measure of the observation is atomic, and the map from f to is bijective, it is shown that there exists an estimator f_n such that for every density f _n E [ |f_n -f | ]=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.

Tasks

Reproductions