Resampling-based confidence regions and multiple tests for a correlated random vector
Sylvain Arlot, Gilles Blanchard and Etienne Roquain
In: 20th conference on Learning Theory (COLT 2007), 13-15 June 2007, San Diego, CA, USA.
We study generalized bootstrapped confidence regions for the mean of a random vector
whose coordinates have an unknown dependence structure, with
a non-asymptotic control of the confidence level. The random
vector is supposed to be either Gaussian or to have a symmetric
%, and we
%observe $n$ i.i.d copies of it.
%The confidence regions are built
%using a data-dependent threshold based on a weighted bootstrap
We consider two approaches, the first based on a
concentration principle and the second on a direct boostrapped
quantile. The first one allows us to deal with
a very large class of resampling weights while our results for
the second are restricted to Rademacher weights. However,
the second method seems more accurate in practice. Our results
are motivated by multiple testing problems, and we show on
simulations that our procedures are better than the
Bonferroni procedure (union bound) as soon as the observed vector has
sufficiently correlated coordinates.