Structural adaptive deconvolution under $L_p$-losses

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.