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Testing relevant hypotheses in functional time series via self‐normalization

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  • Holger Dette
  • Kevin Kokot
  • Stanislav Volgushev

Abstract

We develop methodology for testing relevant hypotheses about functional time series in a tuning‐free way. Instead of testing for exact equality, e.g. for the equality of two mean functions from two independent time series, we propose to test the null hypothesis of no relevant deviation. In the two‐sample problem this means that an L2‐distance between the two mean functions is smaller than a prespecified threshold. For such hypotheses self‐normalization, which was introduced in 2010 by Shao, and Shao and Zhang and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self‐normalized procedures for testing relevant hypotheses in the one‐sample, two‐sample and change point problem and investigate their asymptotic properties. Finite sample properties of the tests proposed are illustrated by means of a simulation study and data examples. Our main focus is on functional time series, but extensions to other settings are also briefly discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:82:y:2020:i:3:p:629-660
    DOI: 10.1111/rssb.12370
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    References listed on IDEAS

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    Cited by:

    1. Holger Dette & Martin Schumann, 2023. "Testing for equivalence of pre-trends in Difference-in-Differences estimation," Papers 2310.15796, arXiv.org.
    2. Hong, Yongmiao & Linton, Oliver & McCabe, Brendan & Sun, Jiajing & Wang, Shouyang, 2024. "Kolmogorov–Smirnov type testing for structural breaks: A new adjusted-range based self-normalization approach," Journal of Econometrics, Elsevier, vol. 238(2).
    3. Kathrin Bissantz & Nicolai Bissantz & Katharina Proksch, 2021. "Nonparametric detection of changes over time in image data from fluorescence microscopy of living cells," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(3), pages 1001-1017, September.
    4. Nick Kloodt & Natalie Neumeyer & Ingrid Keilegom, 2021. "Specification testing in semi-parametric transformation models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(4), pages 980-1003, December.
    5. 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.

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