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Maxisets for density estimation on R

By F. Autin


The problem of density estimation on R is considered. Adopting the maxiset point of view, we focus on performance of adaptive procedures. Any rule which consists in neglecting the wavelet empirical coefficients smaller than a sequence of thresholds vn will be called an elitist rule. We prove that for such a procedure the maximal space for the rate v αp n, with 0 < α < 1, is always contained in the intersection of a Besov space and a weak Besov space. With no assumption on compactness of the support of the density goal f, we show that the hard thresholding rule is the best procedure among elitist rules when taking the classical choice of thresholds vn √ = µ n−1 log(n), with µ> 0. Then, we point out the significance of data-driven thresholds in density estimation by comparing the maxiset of the hard thresholding rule with the one of Juditsky and Lambert-Lacroix’s procedure

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Year: 2006
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