Adaptive thresholding of wavelet coefficients

Wavelet techniques have become an attractive and efficient tool in function estimation. Given noisy data, its discrete wavelet transform is an estimator of the wavelet coefficients. It has been shown by Donoho and Johnstone (Biometrika 81 (1994) 425-455) that thresholding the estimated coefficients and then reconstructing an estimated function reduces the expected risk close to the possible minimum. They offered a global threshold lambda similar to sigma root 2logn for j>j(0), while the coefficients of the first coarse j(0) levels are always included. We demonstrate that the choice of j(0) may strongly affect the corresponding estimators. Then, we use the connection between thresholding and hypotheses testing to construct a thresholding procedure based on the false discovery rate (FDR) approach to multiple testing of Benjamini and Hochberg (J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300). The suggested procedure controls the expected proportion of incorrectly included coefficients among those chosen for the wavelet reconstruction. The resulting procedure is inherently adaptive, and responds to the complexity of the estimated function and to the noise level. Finally, comparing the proposed FDR based procedure with the fixed global threshold by evaluating the relative mean-square-error across the various test-functions and noise levels, we find the FDR-estimator to enjoy robustness of MSE-efficiency.