Some nonasymptotic results on resampling in high dimension, I: Confidence regions
Sylvain Arlot, Gilles Blanchard, and Etienne Roquain
Source: Ann. Statist. Volume 38, Number 1
(2010), 51-82.
Abstract
We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities.
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Keywords: Confidence regions; high-dimensional data; nonasymptotic error control; resampling; cross-validation; concentration inequalities; resampled quantile
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1262271609
Digital Object Identifier: doi:10.1214/08-AOS667
Zentralblatt MATH identifier: 1180.62066
Mathematical Reviews number (MathSciNet): MR2589316
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