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October 2009 Deconvolution with unknown error distribution
Jan Johannes
Ann. Statist. 37(5A): 2301-2323 (October 2009). DOI: 10.1214/08-AOS652

Abstract

We consider the problem of estimating a density fX using a sample Y1, …, Yn from fY=fXfε, where fε is an unknown density. We assume that an additional sample ε1, …, εm from fε is observed. Estimators of fX and its derivatives are constructed by using nonparametric estimators of fY and fε and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density fε, where it is assumed that fX satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density fX belongs to a Sobolev space $H_{\mathcal{p}}$ and fε is ordinary smooth or supersmooth.

Citation

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Jan Johannes. "Deconvolution with unknown error distribution." Ann. Statist. 37 (5A) 2301 - 2323, October 2009. https://doi.org/10.1214/08-AOS652

Information

Published: October 2009
First available in Project Euclid: 15 July 2009

zbMATH: 1173.62018
MathSciNet: MR2543693
Digital Object Identifier: 10.1214/08-AOS652

Subjects:
Primary: 62G07, 62G07
Secondary: 42A38, 62G05

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 5A • October 2009
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