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We consider the non-parametric Poisson regression problem where the integervalued response $Y$ is the realization of a Poisson random variable withparameter $\lambda(X)$. 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 finitedimensional subspaces under mild conditions. In the case of Sobolev ellipsoidsthe obtained rates of convergence turn out to be optimal.  The main part of the paper is devoted to the construction of adaptiveprojection estimators of $\lambda$ via model selection. We proceed in twosteps: first, we assume that an upper bound for $\Vert \lambda \Vert_\infty$ isknown. Under this assumption, we construct an adaptive estimator whosedimension parameter is defined as the minimizer of a penalized contrastcriterion. 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 minimaxoptimal rate up to an additional logarithmic factor both in the independent andthe weakly dependent setup. Appropriate concentration inequalities for Poissonpoint processes turn out to be an important ingredient of the proofs.  We illustrate our theoretical findings by a short simulation study andconclude by indicating directions of future research.

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