# Anisotropic functional deconvolution with long-memory noise: the case of a multi-parameter fractional Wiener sheet

## TL;DR

This paper develops an adaptive wavelet-based method for anisotropic two-dimensional deconvolution with long-memory fractional Gaussian noise, achieving near-optimal convergence rates and extending to higher dimensions without curse of dimensionality.

## Contribution

It introduces a wavelet-vaguelette approach for anisotropic fractional Gaussian noise deconvolution, providing asymptotic optimality and extending results to higher dimensions.

## Key findings

- The estimator attains asymptotically quasi-optimal convergence rates.
- Convergence rates depend on Besov parameters, ill-posedness, and noise parameters.
- Simulation studies support theoretical results.

## Abstract

We look into the minimax results for the anisotropic two-dimensional functional deconvolution model with the two-parameter fractional Gaussian noise. We derive the lower bounds for the $L^p$-risk, $1 \leq p < \infty$, and taking advantage of the Riesz poly-potential, we apply a wavelet-vaguelette expansion to de-correlate the anisotropic fractional Gaussian noise. We construct an adaptive wavelet hard-thresholding estimator that attains asymptotically quasi-optimal convergence rates in a wide range of Besov balls. Such convergence rates depend on a delicate balance between the parameters of the Besov balls, the degree of ill-posedness of the convolution operator and the parameters of the fractional Gaussian noise. A limited simulations study confirms theoretical claims of the paper. The proposed approach is extended to the general $r$-dimensional case, with $r> 2$, and the corresponding convergence rates do not suffer from the curse of dimensionality.