The Annals of Statistics

Fixed-smoothing asymptotics for time series

Xianyang Zhang and Xiaofeng Shao

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In this paper, we derive higher order Edgeworth expansions for the finite sample distributions of the subsampling-based $t$-statistic and the Wald statistic in the Gaussian location model under the so-called fixed-smoothing paradigm. In particular, we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions. The results are used to justify the second-order correctness of a new bootstrap method, the Gaussian dependent bootstrap, in the context of Gaussian location model.

Article information

Ann. Statist., Volume 41, Number 3 (2013), 1329-1349.

First available in Project Euclid: 4 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties

Bootstrap fixed-smoothing asymptotics high-order expansion long-run variance matrix


Zhang, Xianyang; Shao, Xiaofeng. Fixed-smoothing asymptotics for time series. Ann. Statist. 41 (2013), no. 3, 1329--1349. doi:10.1214/13-AOS1113.

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Supplemental materials

  • Supplementary material: Proofs of the other results in Sections 2–3 and simulation results. This supplement contains proofs of the other main results in Sections 2–3 and some simulation results.