Model selection for density estimation with $$\mathbb L _2$$-loss

Abstract

We consider here estimation of an unknown probability density \(s\) belonging to \(\mathbb L _2(\mu )\) where \(\mu \) is a probability measure. We have at hand \(n\) i.i.d. observations with density \(s\) and use the squared \(\mathbb L _2\)-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 \(s\in \mathbb L _2(\mu )\). We shall, in particular, consider the cases of unbounded densities and bounded densities with unknown \(\mathbb L _\infty \)-norm and investigate how the \(\mathbb L _\infty \)-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 \(\mathbb L _2\)-balls with centers belonging to \(\mathbb L _\infty \). Although of a purely theoretical nature, our method leads to results that cannot presently be reached by more concrete ones.