The Annals of Statistics

Deconvolution with unknown error distribution

Jan Johannes

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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.

Article information

Ann. Statist., Volume 37, Number 5A (2009), 2301-2323.

First available in Project Euclid: 15 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G07: Density estimation
Secondary: 62G05: Estimation 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Deconvolution Fourier transform kernel estimation spectral cut off Sobolev space source condition optimal rate of convergence


Johannes, Jan. Deconvolution with unknown error distribution. Ann. Statist. 37 (2009), no. 5A, 2301--2323. doi:10.1214/08-AOS652.

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