On the minimax optimality of block thresholded wavelet estimators

Abstract

Block thresholding methods have been proposed by Hall, Kerkyacharian and Picard (1995) as a means of obtaining increased adaptivity when estimating a function using wavelet methods. For example, it has been shown that block thresholding reduces mean squared error by rendering the estimator more adaptive to relatively subtle, local changes in curvature, of the type that local bandwidth choice is designed to accommodate in traditional kernel methods. In this paper we show that block thresholding also provides extensive adaptivity to many varieties of aberration, including those of chirp and Doppler type. Indeed, in a wide variety of function classes, block thresholding methods possess minimax-optimal convergence rates, and in particular enjoy those rates without the extraneous logarithmic penalties that are usually suffered by term-by-term thresholding methods.