Wavelet block thresholding for samples with random design: a minimax approach under the Lp risk

In recent years, wavelet thresholding procedures have been widely applied to the field of nonparametric function estimation. They excel in the areas of spatial adaptivity, computational efficiency and asymptotic optimality. Among the various thresholding techniques studied in the literature, there are the term-byterm thresholding (hard, soft, . . . ) initially developed by Donoho and Johnstone (1995) and the block thresholding (global, BlockShrink, . . . ) introduced by Kerkyacharian et al. (1996) and Hall et al. (1999). Several recent works demonstrated that the block thresholding methods can enjoy better theoretical properties than the conventional term-by-term thresholding methods. This superiority has been proved for various statistical models via the minimax approach under the L risk. See, for instance, Cai (1999) and Cavalier and Tsybakov (2001) for the Gaussian sequence model, Cai and Chicken (2005) for the density estimation, Chicken (2003) for the regression model with nonequispaced samples and Chicken (2007) for the regression model with random uniform design. This paper presents an extension of a result established by Chicken (2007). We prove that a generalized version of the BlockShrink construction achieves better rates of convergence than the conventional term-by-term thresholding estimators.