Related papers: Consistent Density Estimation Under Discrete Mixture Models

Consistent Density Estimation Under Discrete Mixture Models

URL: http://arxiv.org/abs/2105.01108v1
Date: Mon, 3 May 2021 18:30:02 GMT
Title: Consistent Density Estimation Under Discrete Mixture Models
Authors: Luc Devroye and Alex Dytso
Abstract summary: This work considers a problem of estimating a mixing probability density $f$ in the setting of discrete mixture models. In particular, it is shown that there exists an estimator $f_n$ such that for every density $f$ $lim_nto infty mathbbE left[ int |f_n -f | right]=0$.
Score: 20.935152220339056
License: http://creativecommons.org/licenses/by/4.0/
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 $f$.

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