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.