This paper studies wavelet estimations for supersmooth density functions with additive noises. We first show lower bounds of Lprisk(1 p < ∞) with both moderately and severely ill-posed noises. Then a Shannon wavelet estimator provides optimal or nearly-optimal estimations over Lprisks 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).
The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Reginska＇s work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.
Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.
This paper studies directional Hlder regularity of two-variable functions by their shearlet coefficients, where the shearlets are defined by Guo and Labate(2013). We provide necessary conditions for a function possessing some directional H¨older regularity and the corresponding sufficient conditions, motivated by the work of Sampo and Sumetkijakan(2009) and Lakhonchai et al.(2010).
We define a wavelet linear estimator for density derivative in Besov space based on a negatively associated stratified size-biased random sample. We provide two upper bounds of wavelet estimations on L^p （1 ≤ p 〈 ∞） risk.
