Optimal Wavelet Estimations For A Density With Additive Noises

The density estimation with additive noises plays important roles in medicalstatistics, statistics in sports, astronomy and econometrics. During the last twodecades, wavelets attract lots of attentions as an efective tool and are success-fully applied to density estimations. Motivated by the work of Donoho, Walter,Pensky, Fan, Lounici and etc, this dissertation constructs some estimators forthose densities in Besov and supersmooth spaces with noises respectively, andstudies the convergence rates as well as the optimality of those estimators in theLp(1≤p≤∞) sense.Firstly, we give Lprisk estimations for linear wavelet estimators in Besovspaces. It turns out that those estimators are adaptive under the severely ill-posed noise. Because the estimations are not adaptive with moderately ill-posednoise, we provide non-linear wavelet estimations in the later case.Secondly, the optimal analysis is discussed for above estimators in Besovspaces. More precisely, by considering the lower bound of Lp(1≤p≤∞) riskbetween the density in Besov spaces Bsr,q(R) and any estimators, we prove thefollowing conclusions: For moderately ill-posed noises, the linear wavelet estimatoris optimal when r≥p; the non-linear one attains sub-optimal (which meansoptimal up to a ln n factor) when r <p; Although the linear wavelet estimatordoesn’t provide optimal (sub-optimal) estimations among all estimators for r≤p,it does among all linear estimators; For severely ill-posed noises, the practicallinear wavelet estimator attains optimal.Based on the observation that the optimal convergence rates in Besov spaces are much slower for severely ill-posed noises than that for moderately ill-posedones, we study the convergence rates of Lprisk in supersmooth spaces: TheShannon wavelet estimator is defned and the convergence rates is provided inLp(1<p <∞) sense. Since the Shannon wavelet doesn’t belong to L(R), weconstruct Meyer wavelet estimator and discuss the convergence rates and practi-cability.Finally, we show a lower bound of Lprisk for densities in supersmooth spaces.It turns out that under the above two types of noises, the Shannon wavelet estima-tor attains optimal under moderately ill-posed noises, sub-optimal under severelyill-posed ones for p≥2, and sub-optimal for1≤p <2. In the later case, the ratiosof upper and lower bounds are determined. When p=1and the noises are moder-ately ill-posed, the Meyer wavelet estimator is sub-optimal. An unsolved problemis to provide optimal (sub-optimal) wavelet estimations, when p=1, p=∞andthe noises are severely ill-posed.