Model selection for Poisson processes with covariates
Résumé
We observe $n$ inhomogeneous Poisson processes with covariates and aim at estimating their intensities. We assume that the intensity of each Poisson process is of the form $s (\cdot, x)$ where $x$ is a covariate and where $s$ is an unknown function. We propose a model selection approach where the models are used to approximate the multivariate function $s$. We show that our estimator satisfies an oracle-type inequality under very weak assumptions both on the intensities and the models. By using an Hellinger-type loss, we establish non-asymptotic risk bounds and specify them under several kind of assumptions on the target function $s$ such as being smooth or a product function. Besides, we show that our estimation procedure is robust with respect to these assumptions. This procedure is of theoretical nature but yields results that cannot currently be obtained by more practical ones.