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 Br,qs(L) (r,q⩾1) and over L2L2 risk (Fan and Koo, 2002 [13]). The L∞L∞ risk estimations are investigated by Lounici and Nickl (2011) [19]. This paper deals with optimal estimations over Lp (1⩽p⩽∞)Lp (1⩽p⩽∞) risk for moderately ill-posed noises. A lower bound of LpLp 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⩾pr⩾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.