Electronic Journal of Statistics

Kernel estimates of nonparametric functional autoregression models and their bootstrap approximation

Tingyi Zhu and Dimitris N. Politis

Full-text: Open access


This paper considers a nonparametric functional autoregression model of order one. Existing contributions addressing the problem of functional time series prediction have focused on the linear model and literatures are rather lacking in the context of nonlinear functional time series. In our nonparametric setting, we define the functional version of kernel estimator for the autoregressive operator and develop its asymptotic theory under the assumption of a strong mixing condition on the sample. The results are general in the sense that high-order autoregression can be naturally written as a first-order AR model. In addition, a component-wise bootstrap procedure is proposed that can be used for estimating the distribution of the kernel estimation and its asymptotic validity is theoretically justified. The bootstrap procedure is implemented to construct prediction regions that achieve good coverage rate. A supporting simulation study is presented in the end to illustrate the theoretical advances in the paper.

Article information

Electron. J. Statist., Volume 11, Number 2 (2017), 2876-2906.

Received: October 2016
First available in Project Euclid: 8 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G09: Resampling methods 37M10: Time series analysis 60G25: Prediction theory [See also 62M20]

Functional time series nonparametric autoregression $\alpha$-mixing regression-type bootstrap prediction region

Creative Commons Attribution 4.0 International License.


Zhu, Tingyi; Politis, Dimitris N. Kernel estimates of nonparametric functional autoregression models and their bootstrap approximation. Electron. J. Statist. 11 (2017), no. 2, 2876--2906. doi:10.1214/17-EJS1303. https://projecteuclid.org/euclid.ejs/1502157625

Export citation


  • [1] Antoniadis, A., Paparotidis, E. and Sapatinas, T. (2006). A functional wavelet-kernel approach for time series prediction., Journal of the Royal Statistical Society, Series B. 68 837–857.
  • [2] Antoniadis, A. and Sapatinas, T. (2003). Wavelet methods for continuous time prediction using Hilbert-valued autoregressive processes., Journal of Multivariate Analysis. 87 133–158.
  • [3] Aue, A., Norinho, D. D. and Hörmann, S. (2015). On the prediction of stationary functional time series., Journal of the American Statistical Association. 110 378–392.
  • [4] Bosq, D. (2000)., Linear processes in function space. Springer, New York.
  • [5] Bosq, D. (2007). General linear processes in Hilbert spaces and prediction., Journal of Statistical Planning and Inference. 137 879–894.
  • [6] Delsol, L. (2009) Advances on asymptotic normality in non-parametric functional time series analysis., Statistics: A Journal of Theoretical and Applied Statistics. 43(1) 13–33.
  • [7] Didericksen, D., Kokoszka, P. and Zhang, X. (2012) Empirical properties of forecasts with the functional autoregressive model., Comput. Stat. 27(2) 285–298.
  • [8] Ferraty, F., Keilegom, I. V. and Vieu, P. (2010). On the validity of the bootstrap in non-parametric functional regression., Scandinavian Journal of Statistics. 37 286–306.
  • [9] Ferraty, F., Keilegom, I. V. and Vieu, P. (2012). Regression when both response and predictor are functions., Journal of Multivariate Analysis. 109 10–28.
  • [10] Ferraty, F., Mas, A. and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical aspects., Aust. N. Z. J. Stat. 49 267–286.
  • [11] Ferraty, F. and Vieu, P. (2006)., Nonparametric functional data analysis, Theory and Practice. Springer, New York.
  • [12] Franke, J., Kreiss, J.-P. and Mammen, E. (2002). Bootstrap of kernel smoothing in nonlinear time series., Bernoulli. 8(1) 1–37.
  • [13] Franke, J. and Nyarige, E. (2016). On the residual-based bootstrap for functional autoregression., working paper, Univ. of Kaiserslautern, Germany.
  • [14] Gabrys, R., Horváth, L. and Kokoszka, P. (2010). Tests for error correlation in the functional linear model., Journal of American Statistical Association. 105 1113–1125.
  • [15] Hörmann, S., Kidziński, L. and Hallin, M. (2015). Dynamic functional principal components., Journal of the Royal Statistical Society, Series B. 77(2) 319–348.
  • [16] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data., The Annals of Statistics. 38(3) 1845–1884.
  • [17] Horváth, L. and Kokoszka, P. (2011)., Inference for functional data with applications. Springer Series in Statistics, Springer, New York.
  • [18] Kargin, V. and Onatski, A. (2008). Curve forecasting by functional autoregression., Journal of Multivariate Analysis. 99 2508–2526.
  • [19] Klepsch, J. and Klüppelberg, C. (2017). An innovations algorithm for the prediction of functional linear processes., Journal of Multivariate Analysis. 155 252–271.
  • [20] Klepsch, J., Klüppelberg, C. and Wei, T. (2017). Prediction of functional ARMA processes with an application to traffic data. Econometrics and Statistics., 1 128–149.
  • [21] Kreiss, J.-P. and Lahiri, S. N. (2012). Bootstrap methods for time series., Handbook of Statistics: Time Series Analysis: Methods and Applications. 30(1).
  • [22] Masry, E. (1996). Multivariate regression estimation: Local polynomial fitting for time series., Stochastic Process. Appl. 65 81–101.
  • [23] Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality., Stochastic Process. Appl. 115(1) 155–177.
  • [24] Neumann, M. H. and Kreiss, J.-P. (1998). Regression-type inference in nonparametric autoregression., The Annals of Statistics. 26, 1570–1613.
  • [25] Pan, L. and Politis, D. N. (2016) Bootstrap prediction intervals for linear, nonlinear and nonparametric autoregression (with discussion)., Journal of Statistical Planning and Inference. 177 1–27.
  • [26] Politis, D. N. (2013). Model-free model fitting and predictive distribution (with discussion)., Test. 22(2) 183–250.
  • [27] Politis, D. N. (2015)., Model-free prediction and regression: A transformation-based approach to inference. Springer International Publishing.
  • [28] Politis, D. N. and Romano, J. (1994). Limit theorems for weakly dependent Hilbert space valued random variables with application to the stationary bootstrap., Statistica Sinica. 4 461–476.
  • [29] Ramsay, J. and Silverman, B. W. (1997)., Functional Data Analysis. Springer-Verlag, New York.
  • [30] Ramsay, J. and Silverman, B. W. (2002)., Applied Functional Data Analysis: Methods and Case Studies. Springer-Verlag, New York.
  • [31] Raña, P., Aneiros, G., Vilar, J. and Vieu, P. (2016). Bootstrap confidence intervals in functional nonparametric regression under dependence., Electronic Journal of Statistics. 10 1973–1999.
  • [32] Robinson, P. M. (1983). Nonparametric estimators for time series., J. Time Ser. Anal. 4 185–207.
  • [33] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition., Proc. Nat. Acad. Sci U.S.A. 42 43–47.