Supersmooth density estimations over L p risk by wavelets

This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of L-p risk (1 <= p < infinity) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over L-p risks for p >= 2, and a nearly-optimal result for 1 < p < 2 under both noises. In the nearly-optimal cases, the ratios of upper and lower bounds are determined. When p = 1, we give a nearly-optimal estimation with moderately ill-posed noise by using the Meyer wavelet. Finally, the practical estimators are considered. Our results are motivated by the work of Pensky and Vidakovic (1999), Butucea and Tsybakov (2008), Comte et al. (2006), Lacour (2006) and Lounici and Nickl (2011).