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

An estimator of a delta contaminated mixing density function g(?) of the intensity ? of the Poisson distribution is constructed. The estimator is based on an expansion of the continuous portion g0(?) 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. Conditions on the dictionary and the unknown mixing density that yield a sharp oracle inequality for the norm of the difference between g0(?) and its estimator are formulated. Numerical simulations and comparisons with other recently constructed procedures are shown