It is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions $\mathscr{F}$ and based on a probability measure $P$ holds a.s. if and only if $\mathscr{F} \in \mathrm{CLT}(P)$ and $\int F^2 dP < \infty$, where $F = \sup_{f \in \mathscr{F}}|f|$, and it holds in probability if and only if $\mathscr{F} \in \mathrm{CLT}(P)$. Thus, for a large class of statistics, no local uniformity of the CLT (about $P$) is needed for the bootstrap to work. Consistency of the bootstrap (the bootstrapped law of large numbers) is also characterized. (These results are proved under certain weak measurability assumptions on $\mathscr{F}$.)
Publié le : 1990-04-14
Classification:
Bootstrapping,
empirical processes,
central limit theorem,
law of large numbers,
60F17,
62E20,
60B12
@article{1176990862,
author = {Gine, Evarist and Zinn, Joel},
title = {Bootstrapping General Empirical Measures},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 851-869},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990862}
}
Gine, Evarist; Zinn, Joel. Bootstrapping General Empirical Measures. Ann. Probab., Tome 18 (1990) no. 4, pp. 851-869. http://gdmltest.u-ga.fr/item/1176990862/