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.