The Annals of Statistics

Blockwise Bootstrapped Empirical Process for Stationary Sequences

Peter Buhlmann

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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

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

First available in Project Euclid: 11 April 2007

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Zentralblatt MATH identifier


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

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


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

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