DOI: 10.1093/biomet/asad050 ISSN:

More Efficient Exact Group Invariance Testing: using a Representative Subgroup

N W Koning, J Hemerik
  • Applied Mathematics
  • Statistics, Probability and Uncertainty
  • General Agricultural and Biological Sciences
  • Agricultural and Biological Sciences (miscellaneous)
  • General Mathematics
  • Statistics and Probability


We consider testing invariance of a distribution under an algebraic group of transformations, such as permutations or sign-flips. As such groups are typically huge, tests based on the full group are often computationally infeasible. Hence, it is standard practice to use a random subset of transformations. We improve upon this by replacing the random subset with a strategically chosen, fixed subgroup of transformations. In a generalized location model, we show that the resulting tests are often consistent for lower signal-to-noise ratios. Moreover, we establish an analogy between the power improvement and switching from a t-test to a Z-test under normality. Importantly, in permutation-based multiple testing, the efficiency gain with our approach can be huge, since we attain the same power with much fewer permutations.

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