Abstract

We consider here estimation of an unknown probability density s belonging to L2(mu) where mu is a probability measure. We have at hand n i.i.d. observations with density s and use the squared L2-norm as our loss function. Much has been proved about the risk of various types of estimators (projection estimators, kernel estimators, estimators based on model selection, etc.) when s is a bounded density with a known L_{\infty}-norm \|s\|, in which case risk bounds often depend on \|s\|{\infty}, with a few exceptions like estimation using regular histograms or estimation of densities belonging to some specific Besov spaces as shown by Raynaud-Bouret, Rivoirard and Tuleau-Malot (2011). Here we do not want to put any restriction on s, therefore considering also unbounded densities or bounded densities with unknown L_{\infty}-norm.

We shall deal with estimation by model selection, allowing arbitrary families of finite-dimensional (possibly non-linear) models, with applications to adaptive estimation and estimator selection. When s \in L_{\infty} but \|s\| is unknown, we recover the results corresponding to a known value of \|s\|{\infty}. Although of a purely theoretical nature (the resulting estimator cannot be explicitly computed), our method nevertheless leads to results that are presently not reachable by more concrete methods.