UNDERSTANDING WAVESHRINK - VARIANCE AND BIAS ESTIMATION

Donoho & Johnstone's WaveShrink procedure has proved valuable for func tion estimation and nonparametric regression. WaveShrink is based on t he 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 pa per, we derive computationally efficient formulae for computing the ex act bias, variance and L(2) risk of WaveShrink estimates in finite sam ple situations. We use these formulae to understand the behaviour of W aveShrink estimators; construct approximate confidence intervals and b ias estimates for WaveShrink; and compute ideal thresholds for a given function. We show that hard shrinkage has smaller bias but larger var iance than soft shrinkage, and that significantly smaller thresholds s hould be used for soft shrinkage. We also compute minimax thresholds f or WaveShrink estimators and demonstrate that the minimax thresholds c an nearly achieve the ideal rank for a range of functions.