WAVESHRINK WITH FIRM SHRINKAGE

Abstract

Abstract: Donoho and Johnstone’s (1994) WaveShrink procedure has proven valu-able 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 by Donoho and Johnstone (1994). We derive minimax thresholds and provide for-mulas for computing the pointwise variance, bias, and risk for WaveShrink with firm shrinkage. We study the properties of the shrinkage functions, and demon-strate that firm shrinkage offers advantages over both hard shrinkage (uniformly smaller risk and less sensitivity to small perturbations in the data) and soft shrink-age (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. 1.