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

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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.

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Acknowledgements

The authors express their appreciation to the reviewers for their meticulous review and constructive feedback, which significantly enhanced the quality and clarity of this manuscript. Fraiman and Moreno supported by grant FCE-1-2019-1-156054, Agencia Nacional de Investigación e Innovación, Uruguay. Ransford supported by grants from NSERC and the Canada Research Chairs program.

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Authors and Affiliations

  1. Centro de Matemática, Facultad de Ciencias, Universidad de la República, Montevideo, Uruguay

    Ricardo Fraiman

  2. Instituto de Estadística, Departamento de Métodos Cuantitativos, FCEA, Universidad de la República, Montevideo, Uruguay

    Leonardo Moreno

  3. Département de Mathématiques et de Statistique, Université Laval, Québec City, QC, G1V 0A6, Canada

    Thomas Ransford

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  1. Ricardo Fraiman
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  2. Leonardo Moreno
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  3. Thomas Ransford
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Correspondence to Thomas Ransford.

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Fraiman, R., Moreno, L. & Ransford, T. Application of the Cramér–Wold theorem to testing for invariance under group actions. TEST 33, 379–399 (2024). https://doi.org/10.1007/s11749-023-00899-2

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