Statistica Sinica 18(2008), 1007-1024

WAVELET ESTIMATION VIA BLOCK THRESHOLDING:

A MINIMAX STUDY UNDER $L^p$ RISK

Christophe Chesneau

LMNO

Abstract: We investigate the asymptotic minimax properties of an adaptive wavelet block thresholding estimator under the ${L}^p$ risk over Besov balls. It can be viewed as a $L^p$ version of the BlockShrink estimator developed by Cai (1999, 2002). First we show that it is (near) optimal for numerous statistical models, including certain inverse problems. In this statistical context, it achieves better rates of convergence than the hard thresholding estimator introduced by Donoho and Johnstone (1995). We apply this general result to a deconvolution problem.

Key words and phrases: Besov spaces, block thresholding, convolution in Gaussian white noise model, Lp risk, minimax estimation, wavelets.