The Annals of Statistics

On the range of validity of the autoregressive sieve bootstrap

Jens-Peter Kreiss, Efstathios Paparoditis, and Dimitris N. Politis

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Abstract

We explore the limits of the autoregressive (AR) sieve bootstrap, and show that its applicability extends well beyond the realm of linear time series as has been previously thought. In particular, for appropriate statistics, the AR-sieve bootstrap is valid for stationary processes possessing a general Wold-type autoregressive representation with respect to a white noise; in essence, this includes all stationary, purely nondeterministic processes, whose spectral density is everywhere positive. Our main theorem provides a simple and effective tool in assessing whether the AR-sieve bootstrap is asymptotically valid in any given situation. In effect, the large-sample distribution of the statistic in question must only depend on the first and second order moments of the process; prominent examples include the sample mean and the spectral density. As a counterexample, we show how the AR-sieve bootstrap is not always valid for the sample autocovariance even when the underlying process is linear.

Article information

Source
Ann. Statist., Volume 39, Number 4 (2011), 2103-2130.

Dates
First available in Project Euclid: 26 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1319595460

Digital Object Identifier
doi:10.1214/11-AOS900

Mathematical Reviews number (MathSciNet)
MR2893863

Zentralblatt MATH identifier
1227.62067

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis
Secondary: 62G09: Resampling methods

Keywords
Autoregression bootstrap time series

Citation

Kreiss, Jens-Peter; Paparoditis, Efstathios; Politis, Dimitris N. On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 (2011), no. 4, 2103--2130. doi:10.1214/11-AOS900. https://projecteuclid.org/euclid.aos/1319595460


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