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

Ricardo Fraiman (), Leonardo Moreno () and Thomas Ransford ()
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Ricardo Fraiman: Universidad de la República
Leonardo Moreno: Universidad de la República
Thomas Ransford: Université Laval

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2024, vol. 33, issue 2, No 2, 379-399

Abstract: 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.

Keywords: Cramér–Wold theorem; Random projection; Group action; Test for invariance; Exchangeable distribution; Sign-invariant exchangeable distribution; 62G10; 62H15; 60E05; 60E10 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11749-023-00899-2

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