Simultaneous Wavelet Deconvolution in Periodic Setting

The paper proposes a method of deconvolution in a periodic setting which combines two important ideas, the fast wavelet and Fourier transform-based estimation procedure of Johnstone "et al". ["J. Roy. Statist. Soc. Ser. B"<b>66</b> (2004) 547] and the multichannel system technique proposed by Casey and Walnut [<b>"SIAM Rev"</b>. <b>36</b> (1994) 537]. An unknown function is estimated by a wavelet series where the empirical wavelet coefficients are filtered in an adapting non-linear fashion. It is shown theoretically that the estimator achieves optimal convergence rate in a wide range of Besov spaces. The procedure allows to reduce the ill-posedness of the problem especially in the case of non-smooth blurring functions such as boxcar functions: it is proved that additions of extra channels improve convergence rate of the estimator. Theoretical study is supplemented by an extensive set of small-sample simulation experiments demonstrating high-quality performance of the proposed method. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..