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Structural adaptive deconvolution under \({\mathbb{L}_p}\)-losses

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Abstract

In this paper, we address the problem of estimating a multidimensional density f by using indirect observations from the statistical model Y = X + ε. Here, ε 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 \({\mathbb{L}_p}\)-losses when the error ε 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 ε. 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 ∈ [2,+∞]. Furthermore, our estimation procedure adapts automatically to the possible independence structure of f and this allows us to improve significantly the accuracy of estimation.

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  1. Inst. Math. de Marseille, Aix-Marseille Univ., Marseille, France

    G. Rebelles

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Rebelles, G. Structural adaptive deconvolution under \({\mathbb{L}_p}\)-losses. Math. Meth. Stat. 25, 26–53 (2016). https://doi.org/10.3103/S1066530716010026

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