Wavelet estimations for densities and their derivatives with Fourier oscillating noises

By developing the classical kernel method, Delaigle and Meister provide a nice estimation for a density function with some Fourier-oscillating noises over a Sobolev ballWs 2(L) and over L 2 risk (Delaigle and Meister in Stat. Sin. 21:1065-1092, 2011). The current paper extends their theorem to Besov ball Br,q(L) and L p risk with p,q, r ∈ [1,∞] by using wavelet methods. We firstly show a linear wavelet estimation for densities in Br,q(L) over L p risk, motivated by the work of Delaigle and Meister. Our result reduces to their theorem, when p = q = r = 2. Because the linear wavelet estimator is not adaptive, a nonlinear wavelet estimator is then provided. It turns out that the convergence rate is better than the linear one for r ≤ p. In addition, our conclusions contain estimations for density derivatives as well.