In this work we estimate the regression function for Poisson variables, for a deterministic design in $[0,1]$. Our final estimator, which is adaptive to the data, is selected among a collection of maximum likelihood estimators with respect to a penalized empirical Kullback-Leibler risk. We obtain an oracle inequality over the Kullback-Leibler risk for any fixed size $n$ of the design. Moreover, we state an asymptotic lower bound for this risk over Sobolev spaces and prove that our estimator reaches this rate. Hence, the selected estimator is asymptotically minimax over these spaces. We also present numerical experiments, including a strategy to adjust the constants involved in the penalty function.
