Model selection for density estimation with L2-loss

Lucien Birgé

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arXiv · 0808.1416

Model selection for density estimation with L2-loss

Lucien Birgé

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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. 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.

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Lucien Birgé. 2008-08-10. Model selection for density estimation with L2-loss. https://arxiv.org/abs/0808.1416

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