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June, 1994 Blockwise Bootstrapped Empirical Process for Stationary Sequences
Peter Buhlmann
Ann. Statist. 22(2): 995-1012 (June, 1994). DOI: 10.1214/aos/1176325508


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.


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Peter Buhlmann. "Blockwise Bootstrapped Empirical Process for Stationary Sequences." Ann. Statist. 22 (2) 995 - 1012, June, 1994.


Published: June, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0806.62032
MathSciNet: MR1292553
Digital Object Identifier: 10.1214/aos/1176325508

Primary: 62G09
Secondary: 62G20, 62M10

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 2 • June, 1994
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