Consistent Density Estimation Under Discrete Mixture Models

TL;DR

This paper develops a consistent method for estimating mixing densities in discrete mixture models, demonstrating theoretical guarantees and practical implementation, with a case study on Poisson mixtures.

Contribution

It introduces an $L_1$ consistent estimator for mixing densities under certain conditions and shows its feasibility and applicability to Poisson mixture models.

Findings

Existence of an $L_1$ consistent estimator under atomic measure assumptions.

Feasibility of implementing the estimator computationally.

Consistency of estimation in Poisson 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 $\mu$ of the observation is atomic, and the map from $f$ to $\mu$ is bijective, it is shown that there exists an estimator $f_n$ such that for every density $f$ $\lim_{n\to \infty} \mathbb{E} \left[ \int |f_n -f | \right]=0$. 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 $f$.