%0 Journal Article
%@ 0304-4149
%A Dahlhaus, Rainer
%A Neumann, Michael H.
%C Amsterdam
%D 2001
%F heidok:20767
%I Elsevier
%J Stochastic Processes & Their Applications
%K Locally stationary processes; Nonlinear thresholding; Nonparametric curve estimation;  Preperiodogram; Time series; Wavelet estimators
%P 277-308
%R 10.11588/heidok.00020767
%T Locally Adaptive Fitting of Semiparametric Models to Nonstationary Time Series
%U https://archiv.ub.uni-heidelberg.de/volltextserver/20767/
%V 91
%X We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function µ( · ) and a p-dimensional function theta ( · ) = (theta(1)( · ) , ..., theta(p) ( · ))´ that parametrizes the time-varying spectral density ftheta( · ) (lambda). Whereas the mean function is estimated by a usual kernel estimator, each component of theta ( · ) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of theta(i) ( · ), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback-Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of theta(i) ( · ). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.