Asymptotics for TAYLEX and SIMEX estimators in deconvolution of densities

摘要

We deal with deconvolution problems in density estimation. Assume that the data follow a density, which is a convolution of the original density f being of interest with a noise density f_ε. In order to estimate the density f, one usually should know f_ε completely and then uses some technique for deconvolution. In contrast, the so-called TAYLEX and SIMEX methods introduced by Carroll and Hall and Cook and Stefanski, respectively use partial information on f_ε only and correct the naive density estimator towards the deconvoluted one. In the present paper, we assume that we have more and more information on the noise density when the sample size increases. We show that by applying these methods, one can achieve almost optimal rates and optimal rates respectively for densities f belonging to certain Sobolev classes.