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 and @math in -- refers to the so-called standard deconvolution model which attracted attention of a number of researchers. (Note that the standard deconvolution model is typically ill-posed in the sense of Hadamard: the inversion does not depend continuously on the observed data, i.e., small noise in the convolved signal leads to a significant error in the estimation procedure.) After a rather rapid progress in this problem in late eighties--early nineties, authors turned to adaptive wavelet solutions of the problem that are optimal (in the minimax or the maxiset sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls and for a variety of loss functions defining the risk, and under mild conditions on the blurring function (see, e.g., @cite_7 @cite_6 @cite_26 @cite_2 @cite_11 @cite_21 @cite_29 @cite_8 ).
@cite_26 @cite_7 @cite_8 @cite_29 @cite_21 @cite_6 @cite_2 @cite_11
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We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model. We derive minimax lower bounds for the L 2 -risk in the proposed functional deconvolution model when f(·) is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of f(·) that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls. In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings. 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. We consider the problem of estimating the unknown response function in the multichannel deconvolution model with long-range dependent Gaussian or sub-Gaussian errors. We do not limit our consideration to a specific type of long-range dependence rather we assume that the errors should satisfy a general assumption in terms of the smallest and largest eigenvalues of their covariance matrices. We derive minimax lower bounds for the quadratic risk in the proposed multichannel deconvolution model when the response function is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of the response function that is asymptotically optimal (in the minimax sense), or nearoptimal (within a logarithmic factor), in a wide range of Besov balls, for both Gaussian and sub-Gaussian errors. It is shown that the optimal convergence rates depend on the balance between the smoothness parameter of the response function, the kernel parameters of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the total number of channels. Some examples of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation are used to illustrate the application of the theory we developed. The optimal convergence rates and the adaptive estimators we consider extend the ones studied by Pensky and Sapatinas (2009, 2010) for independent and identically distributed Gaussian errors to the case of long-range dependent Gaussian or sub-Gaussian errors. 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 " 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 = p 5, it has Lebesgue measure zero. In this paper we derive two tuning scenarios of WaveD that are valid for all" boxcar convolutions (i.e., when a 2 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 nds that (i) in the worst case scenario, departures from the BA assumption aect 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.