Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model

TL;DR

This paper introduces an adaptive wavelet block thresholding estimator for accurately estimating derivatives of a function from noisy, convoluted multichannel data, achieving near-optimal minimax rates and demonstrating strong practical performance.

Contribution

The paper develops a novel wavelet block thresholding estimator that attains minimax optimality for derivative estimation in heteroscedastic convolution models, with comprehensive theoretical and numerical validation.

Findings

Estimator achieves near-minimax rates over Besov spaces.

Numerical simulations confirm superior practical performance.

Method outperforms existing approaches on test functions.

Abstract

We observe $n$ heteroscedastic stochastic processes $\{Y_v(t)\}_{v}$, where for any $v\in\{1,\ldots,n\}$ and $t \in [0,1]$, $Y_v(t)$ is the convolution product of an unknown function $f$ and a known blurring function $g_v$ corrupted by Gaussian noise. Under an ordinary smoothness assumption on $g_1,\ldots,g_n$, our goal is to estimate the $d$-th derivatives (in weak sense) of $f$ from the observations. We propose an adaptive estimator based on wavelet block thresholding, namely the "BlockJS estimator". Taking the mean integrated squared error (MISE), our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax in the least favorable situation. We also report a comprehensive suite of numerical simulations to support our theoretical findings. The practical performance of our block estimator compares very favorably to existing methods of the literature on a large set of test functions.