Wavelet smoothing of evolutionary spectra by non–linear thresholding

We consider wavelet estimation of the time–dependent (evolutionary) power spectrum of a locally stationary time series. Hereby, wavelets are used to provide an adaptive local smoothing of a short–time periodogram in the time–frequency plane. For this, in contrast to classical nonparametric (linear) approaches we use non–linear thresholding of the empirical wavelet coefficients. We show how these techniques allow for both adaptively reconstructing the local structure in the time–frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which results into a near–optimal L2–minimax rate for the resulting spectral estimator. Our approach is based on a 2–d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time–varying spectrum motivated from mobile radio propagation.