Mathematics – Statistics Theory
Scientific paper
2008-10-27
Annals of Statistics 2008, Vol. 36, No. 5, 2110-2134
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/07-AOS534 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
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.
Hall Peter
Lahiri Soumendra N.
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