ADAPTIVE VARIANCE FUNCTION ESTIMATION IN HETEROSCEDASTIC NONPARAMETRIC REGRESSION

• 成果类型: Article
• 署名作者: Cai, T. Tony; Wang, Lie
• 署名单位: University of Pennsylvania
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/07-AOS509
• 发表日期: 2008
• 页码: 2025-2054
• 关键词: wavelet shrinkage; INEQUALITY

摘要

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.