Nonparametric Deconvolution by Fourier Transformation and Other Related Approaches

Abstract

Nonparametric deconvolution in density estimation and its companion in regression (i.e., nonparametric regression with measurement errors) have broad applications. Many nonparametric deconvolution methods in the errors-in-variables literature are based on kernel estimation. There are also some nonparametric deconvolution methods constructed based on the Fourier Transformation (since-cosine series), splines, wavelet, and other function expansions in specific basis function spaces. We introduce representative methods in the latter type, as well as a low order approximation that can be used when the error distribution is unknown and the error variance is small. Some recent methods in the related image deblurring area are also described. We consider two types of errors in this inverse problem: supersmooth errors and ordinary smooth errors.