Application of the Cram\'er-Wold theorem to testing for invariance under group actions

TL;DR

This paper introduces a new, efficient test for invariance of probability measures under group actions using one-dimensional projections, applicable even in infinite-dimensional spaces, with practical examples and comparisons.

Contribution

It develops a novel projection-based testing procedure for invariance under group actions, extending to infinite-dimensional data and including special cases like exchangeability.

Findings

Test is powerful and computationally efficient.

Applicable to infinite-dimensional and functional data.

Includes special cases like exchangeability.

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 Cram\'er-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.