Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum

We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t j;n depending on sample size n, wavelet basis / and resolution level j. At fine resolution levels (j = 1; 2; :::), we propose t j;n = ff j log n; where fff j g are level-dependent constants and at coarse levels (j AE 1) t j;n = ß p 3 p log n: The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties