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.

我们解决了在一组线性变换的作用下测试概率测度的不变性的问题。我们提出了一种基于一维投影考虑的程序，并使用 Cramér-Wold 定理的变体进行了证明。我们的测试程序功能强大、计算效率高且与维度无关，甚至可以扩展到无限维空间（多元函数数据）的情况。作为特殊情况，它包括可交换性和符号不变可交换性的测试。我们在小型模拟研究中将我们的程序与这些情况下的一些先前建议进行比较。本文最后给出了两个真实数据示例。