Statistics Papers

Document Type

Journal Article

Date of this Version

2008

Publication Source

Annals of Statistics

Volume

36

Issue

5

Start Page

2025

Last Page

2054

DOI

10.1214/07-AOS509

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.

Keywords

Adaptive estimation, nonparametric regression, thresholding, variance function estimation, wavelets

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Date Posted: 27 November 2017

This document has been peer reviewed.