We apply the bootstrap for general stationary observations, proposed by Kunsch, to the empirical process for $p$-dimensional random vectors. It is known that the empirical process in the multivariate case converges weakly to a certain Gaussian process. We show that the bootstrapped empirical process converges weakly to the same Gaussian process almost surely, assuming that the block length $l$ for constructing bootstrap replicates satisfies $l(n) = O(n^{1/2-\varepsilon}), 0 < \varepsilon < \frac{1}{2}$, and $l(n) \rightarrow \infty$. An example where the multivariate setup arises are the robust GM-estimates in an autoregressive model. We prove the asymptotic validity of the bootstrap approximation by showing that the functional associated with the GM-estimates is Frechet-differentiable.
Publié le : 1994-06-14
Classification:
Bootstrap,
empirical process,
Frechet-differentiability,
GM-estimates,
resampling,
stationary and strong-mixing sequences,
weak convergence,
62G09,
62G20,
62M10
@article{1176325508,
author = {Buhlmann, Peter},
title = {Blockwise Bootstrapped Empirical Process for Stationary Sequences},
journal = {Ann. Statist.},
volume = {22},
number = {1},
year = {1994},
pages = { 995-1012},
language = {en},
url = {http://dml.mathdoc.fr/item/1176325508}
}
Buhlmann, Peter. Blockwise Bootstrapped Empirical Process for Stationary Sequences. Ann. Statist., Tome 22 (1994) no. 1, pp. 995-1012. http://gdmltest.u-ga.fr/item/1176325508/