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 ψ and resolution level j . At fine resolution levels ( j = 1, 2, ...) we propose t j , n = α j log n where {α j } 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.