Anisotropic functional Fourier deconvolution with long-memory dependent errors: a minimax study

TL;DR

This paper studies the limits of estimating functions in a deconvolution model affected by long-memory errors, proposing an adaptive wavelet estimator that nearly achieves optimal rates depending on the long-memory strength.

Contribution

It introduces a wavelet-based estimator for anisotropic deconvolution with long-memory errors, achieving near-minimax convergence rates under specific covariance conditions.

Findings

Estimator adapts to the weakest long-memory dependence

Convergence rates deteriorate with increasing long-memory

Estimator attains asymptotic near-optimality

Abstract

We investigate minimax results for the anisotropic functional deconvolution model when observations are affected by the presence of long-memory. Under specific conditions about the covariance matrices of the errors, we follow a standard procedure to construct an adaptive wavelet-based estimator that attains asymptotically near-optimal convergence rates. These rates depend on the parameter associated with the weakest long-range dependence, and deteriorate as the intensity of long-memory increases. This behavior suggests that the estimator adjusts to the best case scenario and that the weakest LM dominates.