Structural Adaptive Deconvolution under Lp-Losses

In this paper, we address the problem of estimating a multidimensional density f by using indirect observations from the statistical model Y = X + epsilon. Here, e is a measurement error independent of the random vector X of interest and having a known density with respect to Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under L-p-losses when the error e 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 epsilon. As a consequence, we getminimax adaptive upper bounds over a large scale of anisotropic Nikolskii classes and we prove that our estimator is asymptotically rate optimal when p is an element of [2,+infinity]. Furthermore, our estimation procedure adapts automatically to the possible independence structure of f and this allows us to improve significantly the accuracy of estimation.