Title: 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.