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Application of the Cramér–Wold theorem to testing for invariance under group actions

Author

Listed:
  • Ricardo Fraiman

    (Universidad de la República)

  • Leonardo Moreno

    (Universidad de la República)

  • Thomas Ransford

    (Université Laval)

Abstract

We address the problem of testing for the invariance of a probability measure under the action of a group of linear transformations. We propose a procedure based on consideration of one-dimensional projections, justified using a variant of the Cramér–Wold theorem. Our test procedure is powerful, computationally efficient, and dimension-independent, extending even to the case of infinite-dimensional spaces (multivariate functional data). It includes, as special cases, tests for exchangeability and sign-invariant exchangeability. We compare our procedure with some previous proposals in these cases, in a small simulation study. The paper concludes with two real-data examples.

Suggested Citation

  • Ricardo Fraiman & Leonardo Moreno & Thomas Ransford, 2024. "Application of the Cramér–Wold theorem to testing for invariance under group actions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 33(2), pages 379-399, June.
    • Handle: RePEc:spr:testjl:v:33:y:2024:i:2:d:10.1007_s11749-023-00899-2
    DOI: 10.1007/s11749-023-00899-2
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    References listed on IDEAS

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    1. Tarik Bahraoui & Jean‐François Quessy, 2022. "Tests of multivariate copula exchangeability based on Lévy measures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1215-1243, September.
    2. Deheuvels, Paul, 1981. "An asymptotic decomposition for multivariate distribution-free tests of independence," Journal of Multivariate Analysis, Elsevier, vol. 11(1), pages 102-113, March.
    3. Juan Antonio Cuesta-Albertos & Ricardo Fraiman & Thomas Ransford, 2007. "A Sharp Form of the Cramér–Wold Theorem," Journal of Theoretical Probability, Springer, vol. 20(2), pages 201-209, June.
    4. Michael Harder & Ulrich Stadtmüller, 2017. "Testing exchangeability of copulas in arbitrary dimension," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(1), pages 40-60, January.
    5. Reza Modarres, 2008. "Tests of Bivariate Exchangeability," International Statistical Review, International Statistical Institute, vol. 76(2), pages 203-213, August.
    6. Christian Genest & Johanna Nešlehová & Jean-François Quessy, 2012. "Tests of symmetry for bivariate copulas," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(4), pages 811-834, August.
    7. Cuevas, Antonio & Fraiman, Ricardo, 2009. "On depth measures and dual statistics. A methodology for dealing with general data," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 753-766, April.
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