  Classical two-sample permutation tests for equality of distributions have
exact size in finite samples, but they fail to control size for testing
equality of parameters that summarize each distribution. This paper proposes
permutation tests for equality of parameters that are estimated at root-$n$ or
slower rates. Our general framework applies to both parametric and
nonparametric models, with two samples or one sample split into two subsamples.
Our tests have correct size asymptotically while preserving exact size in
finite samples when distributions are equal. They have no loss in local
asymptotic power compared to tests that use asymptotic critical values. We
propose confidence sets with correct coverage in large samples that also have
exact coverage in finite samples if distributions are equal up to a
transformation. We apply our theory to four commonly-used hypothesis tests of
nonparametric functions evaluated at a point. Lastly, simulations show good
finite sample properties, and two empirical examples illustrate our tests in
practice.
