Multivariate wavelet thresholding in anisotropic function spaces

It is well known that multivariate curve estimation under standard (isotropic) smoothness conditions suffers from the "curse of dimensionality". This is reflected by rates of convergence that deteriorate seriously in standard asymptotic settings. Better rates of convergence than those corresponding to isotropic smoothness priors are possible if the curve to be estimated has different smoothness properties in different directions and the estimation scheme is capable of making use of a lower complexity in some of the directions. We consider typical cases of anisotropic smoothness classes and explore how appropriate wavelet estimators can exploit such restrictions on the curve that require an adaptation to different smoothness properties in different directions. It turns out that nonlinear thresholding with an anisotropic multivariate wavelet basis leads to optimal rates of convergence under smoothness priors of anisotropic type. We derive asymptotic results in the model "signal plus Gaussian white noise", where a decreasing noise level mimics the standard asymptotics with increasing sample size.