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On adaptive wavelet estimation of a class of weighted densities
Fabien Navarro1, 2, Christophe Chesneau1, Jalal M. Fadili2

We investigate the estimation of a weighted density taking the form $g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated distribution function and $w$ is a known (non-negative) weight. Such a class encompasses many examples, including those arising in order statistics or when $g$ is related to the maximum or the minimum of $N$ (random or fixed) independent and identically distributed (\iid) random variables. We here construct a new adaptive non-parametric estimator for $g$ based on a plug-in approach and the wavelets methodology. For a wide class of models, we prove that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with $p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared error) over Besov balls. The theoretical findings are illustrated through several simulations.
1 :  LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 :  GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
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Weighted density – density estimation – plug-in approach – wavelets – block thresholding – reliability – series system – parallel system.