The Annals of Statistics

Blockwise Bootstrapped Empirical Process for Stationary Sequences

Peter Buhlmann

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 22, Number 2 (1994), 995-1012.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325508

Digital Object Identifier
doi:10.1214/aos/1176325508

Mathematical Reviews number (MathSciNet)
MR1292553

Zentralblatt MATH identifier
0806.62032

JSTOR
links.jstor.org

Subjects
Primary: 62G09: Resampling methods
Secondary: 62G20: Asymptotic properties 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Bootstrap empirical process Frechet-differentiability GM-estimates resampling stationary and strong-mixing sequences weak convergence

Citation

Buhlmann, Peter. Blockwise Bootstrapped Empirical Process for Stationary Sequences. Ann. Statist. 22 (1994), no. 2, 995--1012. doi:10.1214/aos/1176325508. https://projecteuclid.org/euclid.aos/1176325508


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