Adaptive and optimal pointwise deconvolution density estimations by wavelets

This paper considers multivariate deconvolution density estimations under the local Holder condition by wavelet methods. A pointwise lower bound of the deconvolution model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate. The nonlinear wavelet estimator is introduced for adaptivity, which attains a nearly optimal rate (optimal up to a logarithmic factor). Because the nonlinear wavelet estimator depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data-driven version without any assumption on the estimated functions.