The Annals of Statistics

Estimation of distributions, moments and quantiles in deconvolution problems

Peter Hall and Soumendra N. Lahiri

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 36, Number 5 (2008), 2110-2134.

Dates
First available in Project Euclid: 13 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1223908086

Digital Object Identifier
doi:10.1214/07-AOS534

Mathematical Reviews number (MathSciNet)
MR2458181

Zentralblatt MATH identifier
1148.62028

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62C20: Minimax procedures

Keywords
Bandwidth errors in variables ill-posed problem kernel methods measurement error minimax optimal convergence rate smoothing regularization

Citation

Hall, Peter; Lahiri, Soumendra N. Estimation of distributions, moments and quantiles in deconvolution problems. Ann. Statist. 36 (2008), no. 5, 2110--2134. doi:10.1214/07-AOS534. https://projecteuclid.org/euclid.aos/1223908086


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