Regression function estimation on non compact support in an heteroscesdastic model

We study the problem of nonparametric regression function estimation on non necessarily compact support in a heteroscedastic model with non necessarily bounded variance. A collection of least squares projection estimators on m-dimensional functional linear spaces is built. We prove new risk bounds for the estimator with fixed m and propose a new selection procedure relying on inverse problems methods leading to an adaptive estimator. Contrary to more standard cases, the data-driven dimension is chosen within a random set and the penalty is random. Examples and numerical simulations results show that the procedure is easy to implement and provides satisfactory estimators.