  Permutation tests are widely used in statistics, providing a finite-sample
guarantee on the type I error rate whenever the distribution of the samples
under the null hypothesis is invariant to some rearrangement. Despite its
increasing popularity and empirical success, theoretical properties of the
permutation test, especially its power, have not been fully explored beyond
simple cases. In this paper, we attempt to partly fill this gap by presenting a
general non-asymptotic framework for analyzing the minimax power of the
permutation test. The utility of our proposed framework is illustrated in the
context of two-sample and independence testing under both discrete and
continuous settings. In each setting, we introduce permutation tests based on
U-statistics and study their minimax performance. We also develop exponential
concentration bounds for permuted U-statistics based on a novel coupling idea,
which may be of independent interest. Building on these exponential bounds, we
introduce permutation tests which are adaptive to unknown smoothness parameters
without losing much power. The proposed framework is further illustrated using
more sophisticated test statistics including weighted U-statistics for
multinomial testing and Gaussian kernel-based statistics for density testing.
Finally, we provide some simulation results that further justify the
permutation approach.
