  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.
