More Power by using Fewer Permutations
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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.
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