2008/10/01 by T. Tony Cai, Lie Wang · 2 citations
Economics, Econometrics and Finance · Mathematics · #Advanced Statistical Methods and Models #Financial Risk and Volatility Modeling #Statistical Methods and Inference #math.ST #msc:62G08 #msc:62G20 #stat.TH
paper · pdf · doi:10.1214/07-aos509
published as Annals of Statistics 2008, Vol. 36, No. 5, 2025-2054 · Published in at http://dx.doi.org/10.1214/07-AOS509 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/10/01 · arxiv created 2008/10/27 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.