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Model and estimator selection for density estimation with L2-loss

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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.

This talk is part of the Statistics series.

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