The Annals of Statistics

Sieve bootstrap for functional time series

Efstathios Paparoditis

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A bootstrap procedure for functional time series is proposed which exploits a general vector autoregressive representation of the time series of Fourier coefficients appearing in the Karhunen–Loève expansion of the functional process. A double sieve-type bootstrap method is developed, which avoids the estimation of process operators and generates functional pseudo-time series that appropriately mimics the dependence structure of the functional time series at hand. The method uses a finite set of functional principal components to capture the essential driving parts of the infinite dimensional process and a finite order vector autoregressive process to imitate the temporal dependence structure of the corresponding vector time series of Fourier coefficients. By allowing the number of functional principal components as well as the autoregressive order used to increase to infinity (at some appropriate rate) as the sample size increases, consistency of the functional sieve bootstrap can be established. We demonstrate this by proving a basic bootstrap central limit theorem for functional finite Fourier transforms and by establishing bootstrap validity in the context of a fully functional testing problem. A novel procedure to select the number of functional principal components is introduced while simulations illustrate the good finite sample performance of the new bootstrap method proposed.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3510-3538.

Received: September 2016
Revised: September 2017
First available in Project Euclid: 11 September 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Bootstrap Fourier transform principal components Karhunen–Loève expansion spectral density operator


Paparoditis, Efstathios. Sieve bootstrap for functional time series. Ann. Statist. 46 (2018), no. 6B, 3510--3538. doi:10.1214/17-AOS1667.

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  • Aneiros-Pérez, G., Cao, R. and Vilar-Fernández, J. M. (2011). Functional methods for time series prediction: A nonparametric approach. J. Forecast. 30 377–392.
  • Aue, A., Norinho, D. D. and Hörmann, S. (2015). On the prediction of stationary functional time series. J. Amer. Statist. Assoc. 110 378–392.
  • Bosq, D. (2000). Linear Process in Function Spaces. Springer, Berlin.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Cerovecki, C. and Hörmann, S. (2017). On the CLT for discrete Fourier transforms of functional time series. J. Multivariate Anal. 154 282–295.
  • Cheng, R. and Pourahmadi, M. (1993). Baxter’s inequality and convergence of finite predictors of multivariate stochastic processes. Probab. Theory Related Fields 95 115–124.
  • Dehling, H., Sharipov, O. S. and Wendler, M. (2015). Bootstrap for dependent Hilbert space valued random variables with application to von Mises statistics. J. Multivariate Anal. 133 200–215.
  • Fernández De Castro, B., Guillas, S. and González Manteiga, W. (2005). Functional samples and bootstrap for predicting sulfur dioxide levels. Technometrics 47 212–222.
  • Franke, J. and Nyarigue, E. (2016). Residual-based bootstrap for functional autoregressions. Preprint.
  • Hörmann, S. and Kidziński, L. (2015). A note on estimation in Hilbertian linear models. Scand. J. Stat. 42 43–62.
  • Hörmann, S., Kidziński, L. and Hallin, M. (2015). Dynamic functional principal components. J. Roy. Statist. Soc. Ser. B 77 319–348.
  • Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845–1884.
  • Hörmann, S. and Kokoszka, P. (2012). Functional time series. In Time Series Analysis-Methods and Applications 157–186.
  • Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer, New York.
  • Horváth, L., Kokoszka, P. and Reeder, R. (2013). Estimation of the mean of functional time series and a two sample problem. J. Roy. Statist. Soc. Ser. B 75 103–122.
  • Horváth, L., Kokoszka, P. and Rice, G. (2014). Testing stationarity of functional time series. J. Econometrics 179 66–82.
  • Hurvich, C. M. and Tsai, C.-L. (1993). A corrected Akaike information criterion for vector autoregressive model selection. J. Time Series Anal. 14 271–279.
  • Hyndman, R. J. and Shang, H. L. (2009). Forecasting functional time series. J. Korean Statist. Soc. 38 199–211.
  • Kreiss, J.-P. (1988). Asymptotic statistical inference for a class of stochastic processes. Habilationsschrift, Univ. Hamburg.
  • Kreiss, J.-P. and Paparoditis, E. (2011). Bootstrap methods for dependent data: A review. J. Korean Statist. Soc. 40 357–378.
  • Kreiss, J.-P., Paparoditis, E. and Politis, D. N. (2011). On the range of validity of the autoregressive sieve bootstrap. Ann. Statist. 39 2103–2130.
  • Lahiri, S. N. (2003). Resampling Methods for Dependent Data. Springer, Berlin.
  • Li, Y., Wang, N. and Carorol, R. J. (2013). Selecting the number of principal components in functional data. J. Amer. Statist. Assoc. 108 1284–1294.
  • McMurry, T. and Politis, D. N. (2011). Resampling methods for functional data. In The Oxford Handbook of Functional Data Analysis (F. Ferraty and Y. Romain, eds.) 189–209. Oxford Univ. Press, London.
  • Merlevède, F., Peligrad, M. and Utev, S. (1997). Sharp conditions for the CLT of linear processes in a Hilbert space. J. Theoret. Probab. 10 681–693.
  • Meyer, M., Jentsch, C. and Kreiss, J.-P. (2017). Baxter’s inequality and sieve bootstrap for random fields. Bernoulli 23 2988–3020.
  • Meyer, M. and Kreiss, J.-P. (2015). On the vector autoregressive sieve bootstrap. J. Time Series Anal. 36 377–397.
  • Mingotti, N., Lillo, R. E. and Romo, J. (2015). A random walk test for functional time series. UC3M working papers, Statistics and Econometrics.
  • Panaretos, V. and Tavakoli, S. (2013). Fourier analysis of stationary time series in function spaces. Ann. Statist. 41 568–603.
  • Paparoditis, E. (2018). Supplement to “Sieve bootstrap for functional time series.” DOI:10.1214/17-AOS1667SUPP.
  • Politis, D. N. and Romano, J. (1994). Limit theorems for weakly dependent Hilbert space valued random variables with applications to the stationary bootstrap. Statist. Sinica 4 461–476.
  • Pourahmadi, M. (2001). Foundation of Time Series Analysis and Prediction Theory. Wiley, New York.
  • Shang, L. H. (2016). Resampling methods for dependent functional data. Preprint.
  • Sharipov, O., Tewes, J. and Wendler, M. (2016). Sequential block bootstrap in a Hilbert space with application to change point analysis. Canad. J. Statist. 44 300-322.
  • Wiener, N. and Masani, P. (1958). The prediction theory of multivariate stochastic processes, II. Acta Math. 99 93–137.
  • Yao, F., Müller, H. G. and Wang, J. L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.
  • Zhou, T. and Politis, D. N. (2016). Kernel estimation of first-order nonparametric functional autoregression model and its bootstrap approximation. Preprint.

Supplemental materials

  • Supplement to “Sieve bootstrap for functional time series”. The online supplement contains the proofs that were omitted in this paper and additional numerical results.