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October 2008 Adaptive variance function estimation in heteroscedastic nonparametric regression
T. Tony Cai, Lie Wang
Ann. Statist. 36(5): 2025-2054 (October 2008). DOI: 10.1214/07-AOS509


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.


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T. Tony Cai. Lie Wang. "Adaptive variance function estimation in heteroscedastic nonparametric regression." Ann. Statist. 36 (5) 2025 - 2054, October 2008.


Published: October 2008
First available in Project Euclid: 13 October 2008

zbMATH: 1148.62021
MathSciNet: MR2458178
Digital Object Identifier: 10.1214/07-AOS509

Primary: 62G08 , 62G20

Keywords: adaptive estimation , Nonparametric regression , thresholding , variance function estimation , Wavelets

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 5 • October 2008
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