  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. The purpose of
this paper is to provide an abstract but completely general method for
estimating s by model selection, allowing to handle arbitrary families of
finite-dimensional (possibly non-linear) models and any density s belonging to
L2(mu). We shall, in particular, consider the cases of unbounded densities and
bounded densities with unknown bound and investigate how the L-infinity-norm of
s may influence the risk. We shall also provide applications to adaptive
estimation and aggregation of preliminary estimators. Although of a purely
theoretical nature, our method leads to results that cannot presently be
reached by more concrete methods.
