Density Deconvolution in a Non-standard Case of Heteroscedastic Noises

We study the density deconvolution problem with heteroscedastic noises whose densities are known exactly and Fourier-oscillating. Based on available data, we propose a nonparametric estimator depending on two regularization parameters. This estimator is shown to be consistency with respect to the mean integrated squared error. We then establish upper and lower bounds of the error over the Sobolev class of target density to give the minimax optimality of the estimator. In particular, this estimator is adaptive to the smoothness of the unknown target density. We finally demonstrate that the estimator achieves the minimax rates when the noise densities are supersmooth and ordinary smooth.