On adaptive wavelet estimation of a derivative and other related linear inverse problems

Abstract

We consider a block thresholding and vaguelet–wavelet approach to certain statistical linear inverse problems. Based on an oracle inequality, an adaptive block thresholding estimator for linear inverse problems is proposed and the asymptotic properties of the estimator are investigated. It is shown that the estimator enjoys a higher degree of adaptivity than the standard term-by-term thresholding methods; it attains the exact optimal rates of convergence over a range of Besov classes. The problem of estimating a derivative is considered in more detail as a test for the general estimation procedure. We show that the derivative estimator is spatially adaptive; it automatically adapts to the local smoothness of the function and attains the local adaptive minimax rate for estimating a derivative at a point.