Nonparametric function estimation under Fourier-oscillating noise

Abstract

In the popular deconvolution problem, the goal is to estimate a curve f from data that only allow direct estimation of another curve g, the convolution of f and a so-called error density. Unlike the standard assumption in deconvolution, we consider a more general setting where the characteristic function of the error density can have zeros. This problem is important as the characteristic functions of uniform distributions, and more generally of many compactly supported distributions, have some zeros. We propose a new nonparametric deconvolution estimator, prove that its convergence rates are not affected by the zeros if f has a finite left endpoint, and we show rate-adaptivity. We suggest data-driven bandwidth selectors and examine their finite sample behaviour via simulated examples.