Adaptive Thresholding of Wavelet

Wavelet techniques have become an attractive and eecient tool in function estimation. Given noisy data, its discrete wavelet transform is an estimator of the wavelet coeecients. It has been shown by Donoho and Johnstone (1994) that thresholding the estimated coeecients and then reconstructing an estimated function reduces the expected risk close to the possible minimum. They ooered a global threshold p 2 logn for j > j 0 , while the coeecients of the rst coarse j 0 levels are always included. We demonstrate that the choice of j 0 may strongly aaect 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 (1995). The suggested procedure controls the expected proportion of incorrectly included coeecients 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 xed global threshold by evaluating the relative Mean-Square-Error across the various test-functions and noise levels, we nd the FDR-estimator to enjoy robustness of MSE-eeciency.