Adaptive boxcar deconvolution on full Lebesgue measure setsReport as inadecuate

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1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires 2 MODAL-X - Modélisation aléatoire de Paris X 3 School of Mathematics and statistics Sydney

Abstract : We consider the nonparametric estimation of a function that is observed in white noise after convolution with a boxcar, the indicator of an interval $-a,a$. In a recent paper \citet{jkpr04} have developped a wavelet deconvolution algorithm called {\tt WaveD} that can be used for -certain- boxcar kernels. For example, {\tt WaveD} can be tuned to achieve near optimal rates over Besov spaces when $a$ is a Badly Approximable BA irrational number. While the set of all BA-s contains quadratic irrationals e.g. $a=\sqrt{5}$ it has Lebesgue measure zero, however. In this paper we derive two tuning scenarios of {\tt WaveD} that are valid for -almost all- boxcar convolution i.e. when $a\in A$ where $A$ is a full Lebesgue measure set. We propose i a tuning inspired from Minimax theory over Besov spaces; ii a tuning inspired from Maxiset theory providing similar rates as for BA numbers. Asymptotic theory informs that i in the worst case scenario, departure from the BA assumption, affects {\tt WaveD} convergence rates, at most, by log factors; ii the Maxiset tuning, which yields smaller thresholds, is superior to the Minimax conservative tuning over a whole range of Besov sup-scales. Our asymptotic results are illustrated in an extensive simulation of boxcar convolution observed in white noise.

Keywords : deconvolution non-parametric regression Meyer wavelet Adaptive estimation

Author: Gérard Kerkyacharian - Dominique Picard - Marc Raimondo -



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