Statistica Sinica 17(2007), 317-340

ADAPTIVE BOXCAR DECONVOLUTION

ON FULL LEBESGUE MEASURE SETS

Gérard Kerkyacharian, Dominique Picard and Marc Raimondo

Université de Paris X-Nanterre, Universités de Paris VII

and University of Sydney

Abstract: We consider the non-parametric 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 Johnstone, Kerkyacharian, Picard and Raimondo (2004) have developed a wavelet deconvolution method (called WaveD) that can be used for ``certain'' boxcar kernels. For example, 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. In this paper we derive two tuning scenarios of WaveD that are valid for ``almost all'' boxcar convolutions (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 derived from Maxiset theory providing similar rates as for WaveD in the BA widths setting. Asymptotic theory finds that (i) in the worst case scenario, departures from the BA assumption affect WaveD convergence rates, at most, by log factors; (ii) the Maxiset tuning, which yields smaller thresholds, is superior to the Minimax tuning over a whole range of Besov sub-scales. Our asymptotic results are illustrated in an extensive simulation of boxcar convolution observed in white noise.

Key words and phrases: Adaptive estimation, boxcar, deconvolution, non-parametric regression, Meyer wavelet.