Title

Multichannel deconvolution with long-range dependence: A minimax study

Authors

Authors

R. Benhaddou; R. Kulik; M. Pensky;T. Sapatinas

Comments

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Abbreviated Journal Title

J. Stat. Plan. Infer.

Keywords

Adaptivity Besov spaces; Block thresholding; Deconvolution Fourier; analysis; Functional data; Long-range dependence; Meyer wavelets; Minimax estimators; Multichannel deconvolution; Partial differential; equations; Stationary sequences; Sub-Gaussianity; Wavelet analysis; WAVELET DECONVOLUTION; BOXCAR DECONVOLUTION; INVERSE PROBLEMS; FUNCTIONAL DECONVOLUTION; REGRESSION; ERRORS; DECOMPOSITION; MODEL; Statistics & Probability

Abstract

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 near-optimal (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. (C) 2014 Elsevier B.V. All rights reserved.

Journal Title

Journal of Statistical Planning and Inference

Volume

148

Publication Date

1-1-2014

Document Type

Article

Language

English

First Page

1

Last Page

19

WOS Identifier

WOS:000334133500001

ISSN

0378-3758

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