Asymptotic Invariance and Robustness of Randomization Tests

Proper testing of hypotheses requires adherence to the relevant assumptions on the data and model under consideration. It is of interest to see if specific hypothesis tests are robust to deviations from such assumptions. These topics have been extensively studied in classic parametric hypothesis testing. In turn, this work considers such questions for randomization tests. Specifically, are these nonparametric tests invariant or robust to the breaking of assumptions? In this work, general randomization tests are considered, which randomize data through application of group actions from some appropriately chosen compact topological group with respect to its Haar measure. It is shown that inferences made utilizing group actions coincides with standard distributional approaches. It is also shown that robustness is often asymptotically achievable even if the data does not necessarily satisfy invariance assumptions. Specific hypothesis tests are considered as examples. These are the one-sample location test and the group of reflections, the two-sample test for equality of means and the symmetric group of permutations, and the Durbin-Watson test for serial correlation and the special orthogonal group of n-dimensional rotations.