On Poisson signal estimation under Kullback-Leibler discrepancy and squared risk

Regression problems under Poisson variability arise in many different scientific areas such as, for examples, astrophysics and medical imaging. This article considers the problem of bandwidth selection for kernel smoothing of Poisson data. Its first contribution is the proposal of a new bandwidth selection method that aims to choose the bandwidth that minimizes the Kullback-Leibler (KL) distance between the estimated and the unknown true regression functions. The idea behind is to first construct an estimator of the KL distance and then chooses the minimizer of this distance estimator as the bandwidth. The consistency of this distance estimator is established. As a second contribution, this article establishes the consistency of an existing estimator that targets the L2 risk between the true and the estimated regression functions. In a simulation study, when the targeting distance measure is the KL discrepancy, the proposed KL-based bandwidth selector outperforms a bandwidth selector that uses deviance cross-validation.