Choice of thresholds for wavelet shrinkage estimate of the spectrum

Abstract

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 tj,n depending on sample size n, wavelet basis ψ and resolution level j. At fine resolution levels (j = 1, 2, ...) we propose t j,n = αj log nwhere {αj} are level‐dependent constants and at coarse levels (j≫ 1) t j,n = (π/√3)(log n)1/2.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.