Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere

TL;DR

This paper introduces a needlet-based adaptive thresholding method for nonparametric heteroskedastic variance estimation on the sphere, achieving minimax-optimal convergence rates by leveraging multiresolution analysis.

Contribution

It proposes a novel needlet-based estimator that adapts to unknown smoothness for variance estimation on the sphere, with proven optimal convergence rates.

Findings

Achieves minimax-optimal convergence rates over Besov spaces.

Effectively adapts to unknown smoothness of the variance function.

Utilizes spatial and spectral localization of needlets.

Abstract

This paper investigates the nonparametric estimation of a heteroskedastic variance function on the sphere in a regression framework, assuming the variance belongs to a Besov regularity class. A needlet-based estimator is proposed, combining multiresolution analysis with hard thresholding. The method exploits the spatial and spectral localization of needlets to adapt to unknown smoothness and is shown to attain minimax-optimal convergence rates over Besov spaces.