Wavelet block thresholding for samples with random design: a minimax approach under the $L^p$ risk

TL;DR

This paper introduces an adaptive wavelet block thresholding estimator for regression with random design, demonstrating its near-optimal minimax performance under $\,L^p$ risk and improved convergence rates over traditional methods.

Contribution

It develops a new wavelet block thresholding approach for random design regression, achieving near-minimax optimality and better convergence than existing estimators.

Findings

Estimator is near minimax optimal under $L^p$ risk.

Achieves better convergence rates than traditional term-by-term estimators.

Effective for Besov ball function spaces.

Abstract

We consider the regression model with (known) random design. We investigate the minimax performances of an adaptive wavelet block thresholding estimator under the $\mathbb{L}^p$ risk with $p\ge 2$ over Besov balls. We prove that it is near optimal and that it achieves better rates of convergence than the conventional term-by-term estimators (hard, soft,...).