Open Access
October 2008 Estimation of distributions, moments and quantiles in deconvolution problems
Peter Hall, Soumendra N. Lahiri
Ann. Statist. 36(5): 2110-2134 (October 2008). DOI: 10.1214/07-AOS534


When using the bootstrap in the presence of measurement error, we must first estimate the target distribution function; we cannot directly resample, since we do not have a sample from the target. These and other considerations motivate the development of estimators of distributions, and of related quantities such as moments and quantiles, in errors-in-variables settings. We show that such estimators have curious and unexpected properties. For example, if the distributions of the variable of interest, W, say, and of the observation error are both centered at zero, then the rate of convergence of an estimator of the distribution function of W can be slower at the origin than away from the origin. This is an intrinsic characteristic of the problem, not a quirk of particular estimators; the property holds true for optimal estimators.


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Peter Hall. Soumendra N. Lahiri. "Estimation of distributions, moments and quantiles in deconvolution problems." Ann. Statist. 36 (5) 2110 - 2134, October 2008.


Published: October 2008
First available in Project Euclid: 13 October 2008

zbMATH: 1148.62028
MathSciNet: MR2458181
Digital Object Identifier: 10.1214/07-AOS534

Primary: 62G20
Secondary: 62C20

Keywords: bandwidth , errors in variables , Ill-posed problem , kernel methods , measurement error , minimax , optimal convergence rate , regularization , smoothing

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 5 • October 2008
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