Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra
- Neumann, Michael H.
- von Sachs, Rainer
2010 Mathematics Subject Classification
- 62G07 62M15 62E20 62M10
- Anisotropic smoothness classes, adaptive estimation, optimal rate of convergence, wavelet thresholding, tensor product basis, time-frequency plane, locally stationary time series, evolutionary spectrum
We derive minimax rates for estimation in anisotropic smoothness classes. This rate is 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, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.