Adaptive variance function estimation in heteroscedastic nonparametric regression

TL;DR

This paper introduces a wavelet thresholding method for adaptive estimation of variance functions in heteroscedastic nonparametric regression, achieving near-optimal convergence rates and adaptivity to function smoothness.

Contribution

The paper proposes a data-driven wavelet thresholding estimator for variance functions that adapts to unknown smoothness levels in both mean and variance functions.

Findings

Estimator achieves near-optimal adaptive rate of convergence.

Method is adaptively near minimax optimal under global mean squared error.

Numerical results demonstrate practical effectiveness.

Abstract

We consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a logarithmic factor of the minimax risk under the global mean integrated squared error over a collection of spatially inhomogeneous function classes. Numerical implementation and simulation results are also discussed.