Adaptive variance function estimation in heteroscedastic nonparametric regression

Details Adaptive variance function estimation in heteroscedastic nonparametric regression Adaptive variance function estimation in heteroscedastic nonparametric regression

2008-10-27

arxiv.org/abs/0810.4780v1

Annals of Statistics 2008, Vol. 36, No. 5, 2025-2054

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/07-AOS509 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

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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