Deconvolution Problems in Nonparametric Statistics
Deconvolution problems occur in many ?elds of nonparametric statistics, for example, density estimation based on contaminated data, nonparametric - gression with errors-in-variables, image and signal deblurring. During the last two decades, those topics have received more and more attention. As appli- tions of deconvolution procedures concern many real-life problems in eco- metrics, biometrics, medical statistics, image reconstruction, one can realize an increasing number of applied statisticians who are interested in nonpa- metric deconvolution methods; on the other hand, some deep results from Fourier analysis, functional analysis, and probability theory are required to understand the construction of deconvolution techniques and their properties so that deconvolution is also particularly challenging for mathematicians. Thegeneraldeconvolutionprobleminstatisticscanbedescribedasfollows: Our goal is estimating a function f while any empirical access is restricted to some quantity h = f?G = f(x?y)dG(y), (1. 1) that is, the convolution of f and some probability distribution G. Therefore, f can be estimated from some observations only indirectly. The strategy is ˆ estimating h ?rst; this means producing an empirical version h of h and, then, ˆ applying a deconvolution procedure to h to estimate f. In the mathematical context, we have to invert the convolution operator with G where some reg- ˆ ularization is required to guarantee that h is contained in the invertibility ˆ domain of the convolution operator. The estimator h has to be chosen with respect to the speci?c statistical experiment.
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ˆf(x ˆm(x apply assumption asymptotic bandwidth Berkson bias term Cauchy–Schwarz inequality compactly supported consider consistency const constant contained continuously differentiable convergence rates converges to zero convolution data X1 deconvolution kernel estimator deconvolution problem defined denotes density class density deconvolution density function derive distribution empirical version error density error-free errors-in-variables regression estimator 2.7 exp(ita fft(t finite Fourier series Fourier transform Fubini's theorem function f gft(t grid points Hellinger distance Hence Hölder Hölder condition holds true inequality integral interval ISE(b kernel density estimator kernel function Lebesgue Lebesgue measure Lemma lower bound MISE nonparametric observation optimal Parseval's identity pointspread function probability measure proof Proposition random variables regression function respect restricted ridge-parameter Sect select the bandwidth sequence smooth g smoothness degree Sobolev conditions statistical stochastic sufficiently large target density Theorem A.4 upper bound variance whole real line