We consider testing invariance of a distribution under an algebraic group of transformations, which includes permutations. In this context, it is commonly believed that one should strive to construct a test based on the entire group. We find that one can sometimes obtain dramatically more power by replacing the entire group with a tiny subgroup. Surprisingly, this allows us to obtain much more power at a much lower computational cost. We examine this finding in the popular group invariance-based Westfall & Young MaxT multiple testing method. Studying the relative efficiency in a Gaussian location model, we find the power gain to be largest in high-dimensional settings.

翻译：我们考虑在包含置换的变换代数群下分布不变性的检验问题。在此背景下，普遍认为应构建基于整个群的检验。然而我们发现，将整个群替换为极小子群有时能显著提升检验功效。令人惊讶的是，这种策略能以更低的计算成本获得更强的统计效力。我们在流行的基于群不变性的Westfall & Young MaxT多重检验方法中验证了这一发现。通过高斯位置模型中的相对效率研究，发现高维场景下的功效增益最为显著。