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2013 Estimation of a distribution from data with small measurement errors
Ann-Kathrin Bott, Luc Devroye, Michael Kohler
Electron. J. Statist. 7: 2457-2476 (2013). DOI: 10.1214/13-EJS850


In this paper we study the problem of estimation of a distribution from data that contain small measurement errors. The only assumption on these errors is that the average absolute measurement error converges to zero for sample size tending to infinity with probability one. In particular we do not assume that the measurement errors are independent with expectation zero. Throughout the paper we assume that the distribution, which has to be estimated, has a density with respect to the Lebesgue-Borel measure.

We show that the empirical measure based on the data with measurement error leads to an uniform consistent estimate of the distribution function. Furthermore, we show that in general no estimate is consistent in the total variation sense for all distributions under the above assumptions. However, in case that the average measurement error converges to zero faster than a properly chosen sequence of bandwidths, the total variation error of the distribution estimate corresponding to a kernel density estimate converges to zero for all distributions. In case of a general additive error model we show that this result even holds if only the average measurement error converges to zero. The results are applied in the context of estimation of the density of residuals in a random design regression model, where the residual error is not independent from the predictor.


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Ann-Kathrin Bott. Luc Devroye. Michael Kohler. "Estimation of a distribution from data with small measurement errors." Electron. J. Statist. 7 2457 - 2476, 2013.


Published: 2013
First available in Project Euclid: 2 October 2013

zbMATH: 1293.62068
MathSciNet: MR3117103
Digital Object Identifier: 10.1214/13-EJS850

Primary: 62G05
Secondary: 62G20

Rights: Copyright © 2013 The Institute of Mathematical Statistics and the Bernoulli Society


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