Wavelet Thresholding: Beyond the Gaussian I.I.D. Situation

With this article we first like to a give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based onGaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus ([Da93]) on theevolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.