Wavelet Deconvolution in a Periodic Setting with Long-Range Dependent Errors

TL;DR

This paper develops a wavelet-based deconvolution estimator for periodic models with long-range dependent noise, achieving near-optimal convergence rates and demonstrating its effectiveness through theoretical analysis and numerical experiments.

Contribution

It introduces a novel wavelet deconvolution method tailored for long-range dependent errors in a periodic setting, extending existing techniques to handle strong dependence.

Findings

The estimator attains near-optimal convergence rates under various L_p losses.

Long-range dependence negatively impacts the convergence rate.

Numerical results show the proposed method outperforms naive approaches.

Abstract

In this paper, a hard thresholding wavelet estimator is constructed for a deconvolution model in a periodic setting that has long-range dependent noise. The estimation paradigm is based on a maxiset method that attains a near optimal rate of convergence for a variety of L_p loss functions and a wide variety of Besov spaces in the presence of strong dependence. The effect of long-range dependence is detrimental to the rate of convergence. The method is implemented using a modification of the WaveD-package in R and an extensive numerical study is conducted. The numerical study supplements the theoretical results and compares the LRD estimator with na\"ively using the standard WaveD approach.