Wavelet optimal estimations for a density with some additive noises

摘要

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). The L~∞ risk estimations are investigated by Lounici and Nickl (2011). This paper deals with optimal estimations over L~P (1 ≤ p ≤ ∞) 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). In addition, our non-linear wavelet estimator is adaptive to the indices s, r, q and L.