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Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach

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  • Holger Dette

    (Ruhr-Universität Bochum)

  • Kevin Kokot

    (Ruhr-Universität Bochum)

Abstract

In this paper we propose statistical inference tools for the covariance operators of functional time series in the two sample and change point problem. In contrast to most of the literature, the focus of our approach is not testing the null hypothesis of exact equality of the covariance operators. Instead, we propose to formulate the null hypotheses in the form that “the distance between the operators is small”, where we measure deviations by the sup-norm. We provide powerful bootstrap tests for these type of hypotheses, investigate their asymptotic properties and study their finite sample properties by means of a simulation study.

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  • Holger Dette & Kevin Kokot, 2022. "Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 195-231, April.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:2:d:10.1007_s10463-021-00795-2
    DOI: 10.1007/s10463-021-00795-2
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    References listed on IDEAS

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    1. Guo, Jia & Zhou, Bu & Zhang, Jin-Ting, 2018. "Testing the equality of several covariance functions for functional data: A supremum-norm based test," Computational Statistics & Data Analysis, Elsevier, vol. 124(C), pages 15-26.
    2. Davide Pigoli & John A. D. Aston & Ian L. Dryden & Piercesare Secchi, 2014. "Distances and inference for covariance operators," Biometrika, Biometrika Trust, vol. 101(2), pages 409-422.
    3. Stefan Fremdt & Josef G. Steinebach & Lajos Horváth & Piotr Kokoszka, 2013. "Testing the Equality of Covariance Operators in Functional Samples," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 138-152, March.
    4. David Kraus & Victor M. Panaretos, 2012. "Dispersion operators and resistant second-order functional data analysis," Biometrika, Biometrika Trust, vol. 99(4), pages 813-832.
    5. Dimitrios Pilavakis & Efstathios Paparoditis & Theofanis Sapatinas, 2020. "Testing equality of autocovariance operators for functional time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 571-589, July.
    6. Alexander Aue & Diogo Dubart Norinho & Siegfried Hörmann, 2015. "On the Prediction of Stationary Functional Time Series," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 378-392, March.
    7. Olimjon Sh. Sharipov & Martin Wendler, 2020. "Bootstrapping covariance operators of functional time series," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(3), pages 648-666, July.
    8. Holger Dette & Kevin Kokot & Stanislav Volgushev, 2020. "Testing relevant hypotheses in functional time series via self‐normalization," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 629-660, July.
    9. Panaretos, Victor M. & Kraus, David & Maddocks, John H., 2010. "Second-Order Comparison of Gaussian Random Functions and the Geometry of DNA Minicircles," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 670-682.
    10. A. Aue & G. Rice & O. Sönmez, 2020. "Structural break analysis for spectrum and trace of covariance operators," Environmetrics, John Wiley & Sons, Ltd., vol. 31(1), February.
    11. Graciela Boente & Daniela Rodriguez & Mariela Sued, 2018. "Testing equality between several populations covariance operators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 919-950, August.
    12. E. Paparoditis & T. Sapatinas, 2016. "Bootstrap-based testing of equality of mean functions or equality of covariance operators for functional data," Biometrika, Biometrika Trust, vol. 103(3), pages 727-733.
    13. Alexander Aue & Gregory Rice & Ozan Sönmez, 2018. "Detecting and dating structural breaks in functional data without dimension reduction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(3), pages 509-529, June.
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    1. Cárcamo, Javier & Cuevas, Antonio & Rodríguez, Luis-Alberto, 2024. "A uniform kernel trick for high and infinite-dimensional two-sample problems," Journal of Multivariate Analysis, Elsevier, vol. 202(C).

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