Abstract We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the L p -risk ( 1 ≤ p ∞ ) . Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.
The case where @math for each @math ; (i.e., the case where in the multichannel deconvolution model the errors follow independent standard Brownian motions) was first considered in @cite_16 (extending the results obtained in @cite_2 for the case @math ).
@cite_16 @cite_2
1998219449 2148688551
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" 66 (2004) 547] and the multichannel system technique proposed by Casey and Walnut [ "SIAM Rev" . 36 (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.. Deconvolution problems are naturally represented in the Fourier domain, whereas thresholding in wavelet bases is known to have broad adaptivity properties. We study a method which combines both fast Fourier and fast wavelet transforms and can recover a blurred function observed in white noise with "O"l"n" log ("n")-super-2r steps. In the periodic setting, the method applies to most deconvolution problems, including certain 'boxcar' kernels, which are important as a model of motion blur, but having poor Fourier characteristics. Asymptotic theory informs the choice of tuning parameters and yields adaptivity properties for the method over a wide class of measures of error and classes of function. The method is tested on simulated light detection and ranging data suggested by underwater remote sensing. Both visual and numerical results show an improvement over competing approaches. Finally, the theory behind our estimation paradigm gives a complete characterization of the 'maxiset' of the method: the set of functions where the method attains a near optimal rate of convergence for a variety of "L"-super-"p" loss functions. Copyright 2004 Royal Statistical Society.