Nonparametric Poisson regression from independent and weakly dependent observations by model selection

Abstract: We consider the non-parametric Poisson regression problem where the integer valued response $Y$ is the realization of a Poisson random variable with parameter $\lambda(X)$. The aim is to estimate the functional parameter $\lambda$ from independent or weakly dependent observations $(X_1,Y_1),\ldots,(X_n,Y_n)$ in a random design framework. First we determine upper risk bounds for projection estimators on finite dimensional subspaces under mild conditions. In the case of Sobolev ellipsoids the obtained rates of convergence turn out to be optimal. The main part of the paper is devoted to the construction of adaptive projection estimators of $\lambda$ via model selection. We proceed in two steps: first, we assume that an upper bound for $\Vert \lambda \Vert_\infty$ is known. Under this assumption, we construct an adaptive estimator whose dimension parameter is defined as the minimizer of a penalized contrast criterion. Second, we replace the known upper bound on $\Vert \lambda \Vert_\infty$ by an appropriate plug-in estimator of $\Vert \lambda \Vert_\infty$. The resulting adaptive estimator is shown to attain the minimax optimal rate up to an additional logarithmic factor both in the independent and the weakly dependent setup. Appropriate concentration inequalities for Poisson point processes turn out to be an important ingredient of the proofs. We illustrate our theoretical findings by a short simulation study and conclude by indicating directions of future research.