Statistica Sinica 21 (2011), 1065-1092
doi:10.5705/ss.2009.082
NONPARAMETRIC FUNCTION ESTIMATION
UNDER FOURIER-OSCILLATING
NOISE
Aurore Delaigle and Alexander Meister
University of
Melbourne and
Universität Rostock
Abstract: In the popular deconvolution problem, the
goal is to estimate a curve 
from data that only allow direct
estimation of another curve 
, the convolution of 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 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.
Key words and phrases:
Contaminated data, deconvolution, density estimation,
errors-in-variables regression, kernel smoothing, measurement
errors.