## The Annals of Statistics

### Deconvolution with unknown error distribution

Jan Johannes

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 15 July 2009

https://projecteuclid.org/euclid.aos/1247663756

Digital Object Identifier
doi:10.1214/08-AOS652

Mathematical Reviews number (MathSciNet)
MR2543693

Zentralblatt MATH identifier
1173.62018

#### Citation

Johannes, Jan. Deconvolution with unknown error distribution. Ann. Statist. 37 (2009), no. 5A, 2301--2323. doi:10.1214/08-AOS652. https://projecteuclid.org/euclid.aos/1247663756

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