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Gilles Rebelles

Publications and source records attributed to Gilles Rebelles.

2 recordsLinked to original sources
arXiv2015

Pointwise adaptive estimation of a multivariate density under independence hypothesis

In this paper, we study the problem of pointwise estimation of a multivariate density. We provide a data-driven selection rule from the family of kernel estimators and derive for it a pointwise oracle inequality. Using the latter bound, we show that the proposed estimator is minimax and minimax adaptive over the scale of anisotropic Nikolskii classes. It is important to emphasize that our estimation method adjusts automatically to eventual independence structure of the underlying density. This, in its turn, allows to reduce significantly the influence of the dimension on the accuracy of estimation (curse of dimensionality). The main technical tools used in our considerations are pointwise uniform bounds of empirical processes developed recently in Lepski [Math. Methods Statist. 22 (2013) 83-99].

math.ST↗
arXiv2015

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.

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