%0 Journal Article
%T Structural adaptive deconvolution under ${\mathbb{L}_p}$-losses
%+ Institut de Mathématiques de Marseille (I2M)
%A Rebelles, Gilles
%Z 29 pages
%< avec comité de lecture
%@ 1066-5307
%J Mathematical Methods of Statistics
%I Springer
%V 25
%N 1
%P 26-53
%8 2016-01
%D 2016
%Z 1504.06246
%R 10.3103/S1066530716010026
%K density estimation
%K deconvolution
%K kernel estimator
%K oracle inequality
%K adaptation
%K independence structure
%K concentration inequality
%Z 62G05, 62G20
%Z Mathematics [math]/Statistics [math.ST]Journal articles
%X In this paper, we address the problem of estimating a multidimensional density $f$ by using indirect observations from the statistical model $Y=X+\varepsilon$. Here, $\varepsilon$ is a measurement error independent of the random vector $X$ of interest, and having a known density with respect to the Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under $L_p$-losses when the error $\varepsilon$ has a characteristic function with a polynomial decay. To achieve this goal, we first construct a kernel estimator of $f$ which is fully data driven. Then, we derive for it an oracle inequality under very mild assumptions on the characteristic function of the error $\varepsilon$. As a consequence, we get minimax adaptive upper bounds over a large scale of anisotropic Nikolskii classes and we prove that our estimator is asymptotically rate optimal when $p\in[2,+\infty]$. Furthermore, our estimation procedure adapts automatically to the possible independence structure of $f$ and this allows us to improve significantly the accuracy of estimation.
%G English
%L hal-01309246
%U https://hal.science/hal-01309246
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-