Estimation Of A Delta-Contaminated Density Of A Random Intensity Of Poisson Data

Keywords

Empirical Bayes; Lasso penalty; Mixing density; Poisson distribution

Abstract

In the present paper, we constructed an estimator of a delta contaminated mixing density function g(λ) of an intensity λ of the Poisson distribution. The estimator is based on an expansion of the continuous portion go(λ) of the unknown pdf over an overcomplete dictionary with the recovery of the coefficients obtained as the solution of an optimization problem with Lasso penalty. In order to apply Lasso technique in the, so called, prediction setting where it requires virtually no assumptions on the dictionary and, moreover, to ensure fast convergence of Lasso estimator, we use a novel formulation of the optimization problem based on the inversion of the dictionary elements. We formulate conditions on the dictionary and the unknown mixing density that yield a sharp oracle inequality for the norm of the difference between go(λ) and its estimator and, thus, obtain a smaller error than in a minimax setting. Numerical simulations and comparisons with the Laguerre functions based estimator recently constructed by [8] also show advantages of our procedure. At last, we apply the technique developed in the paper to estimation of a delta contaminated mixing density of the Poisson intensity of the Saturn’s rings data.

Publication Date

1-1-2016

Publication Title

Electronic Journal of Statistics

Volume

10

Issue

1

Number of Pages

683-705

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1214/16-EJS1118

Socpus ID

84963998558 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84963998558

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