Laplace deconvolution in the presence of indirect long-memory data

TL;DR

This paper addresses the challenge of estimating a function from noisy convolution data affected by long-range dependent noise, proposing an adaptive kernel estimator that achieves optimal rates considering the noise's dependence.

Contribution

It introduces an adaptive kernel-based estimator for deconvolution with long-memory noise and establishes its minimax optimality under Sobolev smoothness assumptions.

Findings

Estimator attains minimax optimal rates in the presence of long-range dependence.

Optimal rates deteriorate as the level of long-range dependence increases.

The method adapts to unknown smoothness of the target function.

Abstract

We investigate the problem of estimating a function $f$ based on observations from its noisy convolution when the noise exhibits long-range dependence. We construct an adaptive estimator based on the kernel method, derive minimax lower bound for the $L^2$-risk when $f$ belongs to Sobolev space and show that such estimator attains optimal rates that deteriorate as the LRD worsens.