Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra

Abstract

We derive minimax rates for estimation in anisotropic smoothness classes. These rates are attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present. As an important application we introduce a new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was recently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, that is, how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.