Poisson Inverse Problems

In this paper, we fo us on nonparametri estimators in inverse problems for Poisson pro esses involving the use of wavelet de ompositions. Adopting an adaptive wavelet Galerkin dis retization we nd that our method ombines the well know theoreti al advantages of wavelet-vaguelette de ompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple losed form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [2℄ to the ontext of log-intensity fun tions approximated by wavelet series with the use of the Kullba k-Leibler distan e between two point pro esses, we also present an asymptoti analysis of onvergen e rates that justify our approa h. In order to shed some light on the theoreti al results obtained and to examine the a ura y of our estimates in nite samples we illustrate our method by the analysis of some simulated examples.