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A symmetry test for functional data via the empirical characteristic functional

Author

Listed:
  • Hedvika Ranošová

    (Charles University)

  • Daniel Hlubinka

    (Charles University)

Abstract

A test of central symmetry of a functional random variable is proposed. We use the characterization of symmetry via the characteristic functional, that is, we use the fact that the characteristic functional of a symmetric random function is a real-valued functional. Hence, we construct a Cramér–von Mises-type test of the hypothesis that the imaginary part of the characteristic functional vanishes. The test statistic assumes a relatively simple form if we use a Gaussian measure to construct the test. As the test statistic is a degenerate U-statistic, we must use a wild bootstrap to obtain approximate critical values under the null hypothesis. An alternative approach based on a two-sample test for functional data is compared with our test procedure. Several simulations illustrate the performance of the proposed test.

Suggested Citation

  • Hedvika Ranošová & Daniel Hlubinka, 2025. "A symmetry test for functional data via the empirical characteristic functional," Statistical Papers, Springer, vol. 66(5), pages 1-23, August.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:5:d:10.1007_s00362-025-01716-8
    DOI: 10.1007/s00362-025-01716-8
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    References listed on IDEAS

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    1. Petr Čoupek & Viktor Dolník & Zdeněk Hlávka & Daniel Hlubinka, 2024. "Fourier approach to goodness-of-fit tests for Gaussian random processes," Statistical Papers, Springer, vol. 65(5), pages 2937-2972, July.
    2. Chen, Feifei & Meintanis, Simos G. & Zhu, Lixing, 2019. "On some characterizations and multidimensional criteria for testing homogeneity, symmetry and independence," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 125-144.
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    5. Henze, N. & Klar, B. & Meintanis, S. G., 2003. "Invariant tests for symmetry about an unspecified point based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 275-297, November.
    6. Rainer Dyckerhoff & Christophe Ley & Davy Paindaveine, 2015. "Depth-based runs tests for bivariate central symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(5), pages 917-941, October.
    7. Norbert Henze & María Dolores Jiménez‐Gamero, 2021. "A test for Gaussianity in Hilbert spaces via the empirical characteristic functional," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 406-428, June.
    8. Kundu, Subrata & Majumdar, Suman & Mukherjee, Kanchan, 2000. "Central Limit Theorems revisited," Statistics & Probability Letters, Elsevier, vol. 47(3), pages 265-275, April.
    9. Hlávka, Zdeněk & Hlubinka, Daniel & Koňasová, Kateřina, 2022. "Functional ANOVA based on empirical characteristic functionals," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
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