Abstract: Donoho and Johnstone's (1994) WaveShrink procedure has proven valuable for signal de-noising and non-parametric regression. WaveShrink has very broad asymptotic near-optimality properties. In this paper, we introduce a new shrinkage scheme, firm, which generalizes the hard and soft shrinkage proposed b y Donoho and Johnstone (1994). We derive minimax thresholds and provide formulas for computing the pointwise variance, bias, and risk for WaveShrink with firm shrinkage. We study the properties of the shrinkage functions, and demonstrate that firm shrinkage offers advantages over both hard shrinkage (uniformly smaller risk and less sensitivity to small perturbations in the data) and soft shrinkage (smaller bias and overall L2 risk). Software is provided to reproduce all results in this paper.
Key words and phrases: Bias estimation, firm shrinkage, minimax thresholds, non-parametric regression, signal de-noising, trend estimation, variance estimation, wavelet transform, WaveShrink.