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February 2011 Global uniform risk bounds for wavelet deconvolution estimators
Karim Lounici, Richard Nickl
Ann. Statist. 39(1): 201-231 (February 2011). DOI: 10.1214/10-AOS836


We consider the statistical deconvolution problem where one observes n replications from the model Y = X + ϵ, where X is the unobserved random signal of interest and ϵ is an independent random error with distribution φ. Under weak assumptions on the decay of the Fourier transform of φ, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators fn for the density f of X, where f : ℝ → ℝ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of f if the Fourier transform of φ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if φ decays polynomially. We also analyze the case where f is a “supersmooth”/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density f.


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Karim Lounici. Richard Nickl. "Global uniform risk bounds for wavelet deconvolution estimators." Ann. Statist. 39 (1) 201 - 231, February 2011.


Published: February 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1209.62060
MathSciNet: MR2797844
Digital Object Identifier: 10.1214/10-AOS836

Primary: 62G07
Secondary: 62G15

Keywords: Band-limited wavelets , Confidence band , Rademacher process , sup-norm loss , Vapnik–Chervonenkis class

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 1 • February 2011
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