Skip to main content
SpringerLink
Log in

Detecting deviations from second-order stationarity in locally stationary functional time series

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

A time-domain test for the assumption of second-order stationarity of a functional time series is proposed. The test is based on combining individual cumulative sum tests which are designed to be sensitive to changes in the mean, variance and autocovariance operators, respectively. The combination of their dependent p values relies on a joint-dependent block multiplier bootstrap of the individual test statistics. Conditions under which the proposed combined testing procedure is asymptotically valid under stationarity are provided. A procedure is proposed to automatically choose the block length parameter needed for the construction of the bootstrap. The finite-sample behavior of the proposed test is investigated in Monte Carlo experiments, and an illustration on a real data set is provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Antoniadis, A., Sapatinas, T. (2003). Wavelet methods for continuous time prediction using hilbert-valued autoregressive processes. Journal of Multivariate Analysis, 87, 133–158.

    Article  MathSciNet  Google Scholar 

  • Aston, J. A. D., Kirch, C. (2012). Detecting and estimating changes in dependent functional data. Journal of Multivariate Analysis, 109(Supplement C), 204–220.

    Article  MathSciNet  Google Scholar 

  • Aue, A., van Delft, A. (2017). Testing for stationarity of functional time series in the frequency domain. ArXiv e-prints.

  • Aue, A., Gabrys, R., Horváth, L., Kokoszka, P. (2009). Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis, 100, 2254–2269.

    Article  MathSciNet  Google Scholar 

  • Aue, A., Dubart Nourinho, D., Hörmann, S. (2015). On the prediction of stationary functional time series. Journal of the American Statistical Association, 110, 378–392.

    Article  MathSciNet  Google Scholar 

  • Berkes, I., Gabrys, R., Horvath, L., Kokoszka, P. (2009). Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(5), 927–946.

  • Billingsley, P. (1999). Convergence of probability measures. New York: Wiley.

    Book  Google Scholar 

  • Bosq, D. (2000). Linear processes in function spaces, Vol. 149., Lecture notes in statistics New York: Springer.

    Book  Google Scholar 

  • Bosq, D. (2002). Estimation of mean and covariance operator of autoregressive processes in Banach spaces. Statistical inference for Stochastic Processes, 5, 287–306.

    Article  MathSciNet  Google Scholar 

  • Box, G. E. P., Pierce, D. A. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association, 65(332), 1509–1526.

    Article  MathSciNet  Google Scholar 

  • Brillinger, D. (1981). Time series: Data analysis and theory. San Francisco: Holden Day Inc.

    MATH  Google Scholar 

  • Bücher, A., Kojadinovic, I. (2016). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing. Bernoulli, 22(2), 927–968.

    Article  MathSciNet  Google Scholar 

  • Bücher, A., Kojadinovic, I. (2017). A note on conditional versus joint unconditional weak convergence in bootstrap consistency results. Journal of Theoretical Probability, 1–21.

  • Bücher, A., Fermanian, J.-D., Kojadinovic, I. (2018). Combining cumulative sum change-point detection tests for assessing the stationarity of univariate time series. ArXiv e-prints.

  • Davydov, Y. A., Lifshits, M. A. (1985). Fibering method in some probabilistic problems. Journal of Soviet Mathematics, 31(2), 2796–2858.

    Article  Google Scholar 

  • Dehling, H., Sharipov, O. (2005). Estimation of mean and covariance operator for banach space valued autoregressive processes with independent innovations. Statistical Inference for Stochastic Processes, 8, 137–149.

    Article  MathSciNet  Google Scholar 

  • Dehling, H., Durieu, O., Volny, D. (2009). New techniques for empirical processes of dependent data. Stochastic Processes and their Applications, 119(10), 3699–3718.

    Article  MathSciNet  Google Scholar 

  • Dette, H., Preuß, P., Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing. Journal of the American Statistical Association, 106(495), 1113–1124.

    Article  MathSciNet  Google Scholar 

  • Dwivedi, Y., Subba Rao, S. (2011). A test for second-order stationarity of a time series based on the discrete fourier transform. Journal of Time Series Analysis, 32, 68–91.

    Article  MathSciNet  Google Scholar 

  • Ferraty, F., Vieu, P. (2006). Nonparametric functional data analysis: Theory and practice. New York: Springer.

  • Fisher, R. (1932). Statistical methods for research workers. London: Olivier and Boyd.

    MATH  Google Scholar 

  • Hörmann, S., Kokoszka, P. (2010). Weakly dependent functional data. Annals of Statistics, 38(3), 1845–1884.

    Article  MathSciNet  Google Scholar 

  • Hörmann, S., Kidziński, Ł., Hallin, M. (2015). Dynamic functional principal components. Journal of the Royal Statistical Society: Series B, 77(2), 319–348.

  • Horváth, L., Kokoszka, P. (2012). Inference for functional data with applications. New York: Springer.

    Book  Google Scholar 

  • Horvath, L., Huskova, M., Kokoszka, P. (2010). Testing the stability of the functional autoregressive process. Journal of Multivariate Analysis, 101(2), 352–367.

    Article  MathSciNet  Google Scholar 

  • Hsing, T., Eubank, R. (2015). Theoretical foundations of functional data analysis, with an introduction to linear operators. New York: Wiley.

    Book  Google Scholar 

  • Hyndman, R. J., Shang, H. L. (2009). Forecasting functional time series. Journal of the Korean Statistical Society, 38(3), 199–211.

    Article  MathSciNet  Google Scholar 

  • Janson, S., Kaijser, S. (2015). Higher moments of Banach space valued random variables. Memoirs of the American Mathematical Society, 238(1127), vii+110.

    Article  MathSciNet  Google Scholar 

  • Jentsch, C., Subba Rao, S. (2015). A test for second order stationarity of a multivariate time series. Journal of Econometrics, 185, 124–161.

    Article  MathSciNet  Google Scholar 

  • Jin, L., Wang, S., Wang, H. (2015). A new non-parametric stationarity test of time series in the time domain. Royal Statistical Society, 77, 893–922.

    Article  MathSciNet  Google Scholar 

  • Lee, J., Subba Rao, S. (2017). A note on general quadratic forms of nonstationary stochastic processes. Statistics, 51(5), 949–968.

    Article  MathSciNet  Google Scholar 

  • Ljung, G. M., Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65(2), 297–303.

    Article  Google Scholar 

  • Panaretos, V. M., Tavakoli, S. (2013). Fourier analysis of stationary time series in function space. Annals of Statistics, 41(2), 568–603.

    Article  MathSciNet  Google Scholar 

  • Politis, D. N., White, H. (2004). Automatic block-length selection for the dependent bootstrap. Econometric Reviews, 23(1), 53–70.

    Article  MathSciNet  Google Scholar 

  • Sharipov, O., Tewes, J., Wendler, M. (2016). Sequential block bootstrap in a hilbert space with application to change point analysis. Canadian Journal of Statistics, 44(3), 300–322.

    Article  MathSciNet  Google Scholar 

  • Statulevicius, V., Jakimavicius, D. (1988). Estimates of semiinvariants and centered moments of stochastic processes with mixing. I. Lithuanian Mathematical Journal, 28, 226–238.

    Article  Google Scholar 

  • van Delft, A., Eichler, M. (2018). Locally stationary functional time series. Electronic Journal of Statistics, 12(1), 107–170.

    Article  MathSciNet  Google Scholar 

  • van Delft, A., Bagchi, P., Characiejus, V., Dette, H. (2017). A nonparametric test for stationarity in functional time series. ArXiv e-prints.

  • van der Vaart, A., Wellner, J. (1996). Weak convergence and empirical processes, Vol. 1., Springer series in statistics New York: Springer.

    Book  Google Scholar 

  • Vogt, M. (2012). Nonparametric regression for locally stationary time series. The Annals of Statistics, 40, 2601–2633.

    Article  MathSciNet  Google Scholar 

  • Weidmann, J. (1980). Linear operators in Hilbert spaces, Vol. 68., Graduate texts in mathematics New York: Springer.

    Book  Google Scholar 

Download references

Acknowledgements

Financial support by the Collaborative Research Center “Statistical modeling of non-linear dynamic processes” (SFB 823, Teilprojekt A1, A7 and C1) of the German Research Foundation, by the Ruhr University Research School PLUS, funded by Germany’s Excellence Initiative [DFG GSC 98/3], by the DAAD (German Academic Exchange Service) and the National Institute of General Medical Sciences of the National Institutes of Health (Award Number R01GM107639) is gratefully acknowledged. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Parts of this paper were written when Axel Bücher was a postdoctoral researcher at Ruhr-Universität Bochum and while Florian Heinrichs was visiting the Universidad Autónoma de Madrid. The authors would like to thank the institute, and in particular Antonio Cuevas, for its hospitality.

Author information

Authors and Affiliations

  1. Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225, Düsseldorf, Germany

    Axel Bücher

  2. Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, 44780, Bochum, Germany

    Holger Dette & Florian Heinrichs

Authors
  1. Axel Bücher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Holger Dette
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Florian Heinrichs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Holger Dette.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 437 KB)

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bücher, A., Dette, H. & Heinrichs, F. Detecting deviations from second-order stationarity in locally stationary functional time series. Ann Inst Stat Math 72, 1055–1094 (2020). https://doi.org/10.1007/s10463-019-00721-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-019-00721-7

Keywords

Search

Navigation