Estimation of delta-contaminated density of the random intensity of Poisson data

Abstract

In the present paper, we constructed an estimator of a delta contaminated mixing density function $g(λ)$ of the intensity $λ$ of the Poisson distribution. The estimator is based on an expansion of the continuous portion $g_0(λ)$ of the unknown pdf over an overcomplete dictionary with the recovery of the coefficients obtained as 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 dictionary and, moreover, to ensure fast convergence of Lasso estimator, we use a novel formulation of the optimization problem based on inversion of the dictionary elements. The total estimator of the delta contaminated mixing pdf is obtained using a two-stage iterative procedure. We formulate conditions on the dictionary and the unknown mixing density that yield a sharp oracle inequality for the norm of the difference between $g_0 (λ)$ 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 Comte and Genon-Catalot (2015) 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.