Consistent Density Estimation Under Discrete Mixture Models
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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 μ of the observation is atomic, and the map from f to μ is bijective, it is shown that there exists an estimator f_n such that for every density f lim_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.
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