On Adaptivity Of BlockShrink Wavelet Estimator Over Besov Spaces

Cai(1996b) proposed a wavelet method, BlockShrink, for estimating regression functions of unknown smoothness from noisy data by thresholding empirical wavelet co-eecients in groups rather than individually. The BlockShrink utilizes the information about neighboring wavelet coeecients and thus increases the estimation accuracy of the wavelet coeecients. In the present paper, we ooer insights into the BlockShrink procedure and show that the minimax optimality of the BlockShrink estimators holds broadly over a wide range of Besov classes B p;q (M). We prove that the BlockShrink estimators attain the exact optimal rate of convergence over a wide interval of Besov classes with p 2; and the BlockShrink estimators achieves the optimal convergence rate within a logarithmic factor over the Besov classes with p < 2. We also show that the BlockShrink estimators enjoys a smoothness property: if the underlying function is the zero function, then, with high probability, the BlockShrink is also the zero function. Thus the BlockShrink procedure removes pure noise completely.