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.
REF developed a wavelet thresholding procedure that is adaptive to the smoothness of the mean and variance functions, but the results are difficult to extend to non-orthogonal design matrices.
Adaptive variance function estimation in heteroscedastic nonparametric regression
Annals of Statistics 2008, Vol. 36, No. 5, 2025-2054
Mathematics