ADAPTIVELY WAVELET-SMOOTHED WIGNER ESTIMATES OF EVOLUTIONARY SPECTRA

We consider adaptive estimation of the evolutionary spectrum in the mo del of a locally stationary time series in the time-frequency plane. T he estimate is based on a modification of a time-dependent periodogram (spectogram), which can be considered as a sort of smoothed pseudo-Wi gner estimate. In our approach, the separable time-frequency smoothing kernel (weight) function a's built by using 2-d tensor product wavele ts which allows independent smoothing in both time and frequency. The estimator becomes adaptive to the local structure (and possibly differ ent degree of smoothness) of the evolutionary spectrum, as we use non- linear thresholding of the empirical wavelet coefficients. Unlike trad itional linear smoothing schemes, our estimator attains the usual near -optimal L(2)-minimax rate, even for spectra with regularity being inh omogeneously distributed over the time-frequency plane.