Annals of Statistics

TFT-bootstrap: Resampling time series in the frequency domain to obtain replicates in the time domain

Claudia Kirch and Dimitris N. Politis

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A new time series bootstrap scheme, the time frequency toggle (TFT)-bootstrap, is proposed. Its basic idea is to bootstrap the Fourier coefficients of the observed time series, and then to back-transform them to obtain a bootstrap sample in the time domain. Related previous proposals, such as the “surrogate data” approach, resampled only the phase of the Fourier coefficients and thus had only limited validity. By contrast, we show that the appropriate resampling of phase and magnitude, in addition to some smoothing of Fourier coefficients, yields a bootstrap scheme that mimics the correct second-order moment structure for a large class of time series processes. As a main result we obtain a functional limit theorem for the TFT-bootstrap under a variety of popular ways of frequency domain bootstrapping. Possible applications of the TFT-bootstrap naturally arise in change-point analysis and unit-root testing where statistics are frequently based on functionals of partial sums. Finally, a small simulation study explores the potential of the TFT-bootstrap for small samples showing that for the discussed tests in change-point analysis as well as unit-root testing, it yields better results than the corresponding asymptotic tests if measured by size and power.

Article information

Ann. Statist., Volume 39, Number 3 (2011), 1427-1470.

First available in Project Euclid: 13 May 2011

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Zentralblatt MATH identifier

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

Frequency domain bootstrap functional limit theorem nonlinear processes periodogram ratio statistics spectral density estimation surrogate data change-point analysis unit root testing


Kirch, Claudia; Politis, Dimitris N. TFT-bootstrap: Resampling time series in the frequency domain to obtain replicates in the time domain. Ann. Statist. 39 (2011), no. 3, 1427--1470. doi:10.1214/10-AOS868.

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