Deconvolution from Fourier-oscillating error densities under decay and smoothness restrictions

Abstract

This paper is concerned with deconvolution from error or blurring densities whose Fourier transforms have isolated zeros and show oscillatory behaviour; unlike conventional approaches where the Fourier transform decays about monotonously. We introduce specific estimation procedures based on local polynomial approximation in the Fourier domain. Under combined moment and smoothness conditions, we are able to improve the convergence rates compared to existing methods in density deconvolution. The corresponding minimax theory is derived. In compactly supported models as in signal deconvolution and Berkson regression, nearly optimal rates are achieved under conditions which are significantly weaker than those assumed in earlier papers.