Locally adaptive fitting of semiparametric models to nonstationary time series

We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function [mu](·) and a p-dimensional function [theta](·)=([theta](1)(·),...,[theta](p)(·))' that parametrizes the time-varying spectral density f[theta](·)([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.