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Consistent Density Estimation Under Discrete Mixture Models

3 May 2021
Luc Devroye
Alex Dytso
ArXiv (abs)PDFHTML
Abstract

This work considers a problem of estimating a mixing probability density fff in the setting of discrete mixture models. The paper consists of three parts. The first part focuses on the construction of an L1L_1L1​ consistent estimator of fff. In particular, under the assumptions that the probability measure μ\muμ of the observation is atomic, and the map from fff to μ\muμ is bijective, it is shown that there exists an estimator fnf_nfn​ such that for every density fff lim⁡n→∞E[∫∣fn−f∣]=0\lim_{n\to \infty} \mathbb{E} \left[ \int |f_n -f | \right]=0limn→∞​E[∫∣fn​−f∣]=0. The second part discusses the implementation details. Specifically, it is shown that the consistency for every fff 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 fff.

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