Abstract: In this article we investigate the asymptotic and numerical properties of a class of block thresholding estimators for wavelet regression. We consider the effect of block size on global and local adaptivity and the choice of thresholding constant. The optimal rate of convergence for block thresholding with a given block size is derived for both the global and local estimation. It is shown that there are conflicting requirements on the block size for achieving the global and local adaptivity. We then consider the choice of thresholding constant for a given block size by treating the block thresholding as a hypothesis testing problem. The combined results lead naturally to an optimal choice of block size and thresholding constant. We conclude with a numerical study which compares the finite-sample performance among block thresholding estimators as well as with other wavelet methods.
Key words and phrases: Block thresholding, convergence rate, global adaptivity, local adaptivity, minimax estimation, nonparametric regression, smoothing parameter, wavelets.