April 2021 Density deconvolution under general assumptions on the distribution of measurement errors
Denis Belomestny, Alexander Goldenshluger
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Ann. Statist. 49(2): 615-649 (April 2021). DOI: 10.1214/20-AOS1969


In this paper, we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically, deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper, we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and nonvanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.


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Denis Belomestny. Alexander Goldenshluger. "Density deconvolution under general assumptions on the distribution of measurement errors." Ann. Statist. 49 (2) 615 - 649, April 2021. https://doi.org/10.1214/20-AOS1969


Received: 1 July 2019; Revised: 1 January 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1969

Primary: 62G07 , 62G20

Keywords: Characteristic function , density deconvolution , Density estimation , Laplace transform , lower bounds , minimax risk

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 2 • April 2021
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