Waveshrink with firm shrinkage. (English) Zbl 1067.62529

Gao, Hong-Ye; Bruce, Andrew G.

Stat. Sin. 7, No. 4, 855-874 (1997).

Summary: Donoho and Johnstone’s (1994) WaveShrink procedure has proven valuable for signal de-noising and nonparametric 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 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 \(L_2\) risk). Software is provided to reproduce all results in this paper.

MSC:

62G08

Nonparametric regression and quantile regression

Keywords:

Bias estimation, firm shrinkage, minimax thresholds, non-parametric regression, signal de-noising, trend estimation, variance estimation, wavelet transform, WaveShrink.

Software:

S+WAVELETS; S-PLUS