Understanding WaveShrink: Variance and bias estimation

Donoho & Johnstone's WaveShrink procedure has proved valuable for function estimation and nonparametric regression. WaveShrink is based on the principle of shrinking wavelet coefficients towards zero to remove noise. WaveShrink has very broad asymptotic near-optimality properties and achieves the optimal risk to within a factor of log n. In this paper, we derive computationally efficient formulae for computing the exact bias, variance and L(2) risk of WaveShrink estimates in finite sample situations. We use these formulae to understand the behaviour of WaveShrink estimators; construct approximate confidence intervals and bias estimates for WaveShrink; and compute ideal thresholds for a given function. We show that hard shrinkage has smaller bias but larger variance than soft shrinkage, and that significantly smaller thresholds should be used for soft shrinkage. We also compute minimax thresholds for WaveShrink estimators and demonstrate that the minimax thresholds can nearly achieve the ideal rank for a range of functions.