Model selection for density estimation with L2-loss

Abstract

We consider here estimation of an unknown probability density s belonging to L2(μ) where μ 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 s ∈ L2(μ). We shall, in particular, consider the cases of unbounded densities and bounded densities with unknown L∞-norm and investigate how the L∞-norm of s may influence the risk. We shall also provide applications to adaptive estimation and aggregation of preliminary estimators. One major technical tool of our approach is a proof of the existence of suitable tests between L2-balls with centers belonging to L∞. Although of a purely theoretical nature, our method leads to results that cannot presently be reached by more concrete ones. © 2013 Springer-Verlag Berlin Heidelberg.