Minimax Wavelet Estimation Via Block Thresholding

Wavelet shrinkage methods have been very successful in nonparametric regression. The most commonly used wavelet procedures achieve adaptivity through term-by-term thresholding. The resulting estimators attain the minimax rates of convergence up to a logarithmic factor. In the present paper, we propose a block thresholding method where wavelet coef-cients are thresholded in blocks, rather than individually. We show that the esti-mators produced by the procedure are spatially adaptive and asymptotically optimal both for global and local estimation. The estimator attains the exact optimal rates of convergence for global estimation over a range of function classes of inhomogeneous smoothness. The estimator also achieves optimal local adaptivity for estimating regression functions at a point. Moreover, a simulation study shows that the Block-Shrink estimators yield uniformly better results in terms of the mean squared error than the widely used VisuShrink estimator. The procedure is easy to implement and the computational cost is of order O(n).