Wavelet optimal estimations for a density with some additive noises

Using wavelet methods, Fan and Koo study optimal estimations for a density with some additive noises over a Besov ball B-r,q(s) (L) (r,q >= 1) and over L-2 risk (Fan and Koo, 2002 [13]). The L-infinity risk estimations are investigated by Lounici and Nickl (2011) [19]. This paper deals with optimal estimations over L-P (1 <= p <= infinity) risk for moderately ill-posed noises. A lower bound of L-P risk is firstly provided, which generalizes Fan Koo and Lounici-Nickl's theorems; then we define a linear and non-linear wavelet estimators, motivated by Fan Koo and Pensky-Vidakovic's work. The linear one is rate optimal for r >= p, and the non-linear estimator attains suboptimal (optimal up to a logarithmic factor). These results can be considered as an extension of some theorems of Donoho et al. (1996) [10]. In addition, our non-linear wavelet estimator is adaptive to the indices s, r, q and L. (C) 2013 Elsevier Inc. All rights reserved.