Multichannel Deconvolution with Long-Range Dependence: A Minimax Study

TL;DR

This paper develops minimax optimal methods for estimating responses in multichannel deconvolution models with long-range dependent Gaussian errors, extending previous results to more general dependence structures.

Contribution

It introduces a new adaptive wavelet estimator that achieves near-minimax optimality under broad conditions of long-range dependence and various convolution smoothness.

Findings

Derived minimax lower bounds for quadratic risk.

Proposed an adaptive wavelet estimator with near-optimal convergence rates.

Extended previous results to models with long-range dependent errors.

Abstract

We consider the problem of estimating the unknown response function in the multichannel deconvolution model with long-range dependent 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 larger 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. 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 errors.