Studentized two-sample U-statistic permutation test

Inputs: Two samples of sizes $m,n\ge r$, a measurable order-$(r,r)$ kernel symmetric within each group, two supplied nonnegative measurable variance-estimation routines, and a significance level $\alpha \in (0,1)$. Use the U-statistic in Two-sample order-r U-statistic and first projections and the standard-error pivot $S$ in Studentized two-sample U-statistic and permutation distribution, including its value zero when the estimated standard error is zero. Choose the upper alternative statistic $T=S$, lower alternative statistic $T=-S$, or two-sided statistic $T=\mathrm{\mid }S\mathrm{\mid }$. Optionally supply an independent uniform random variable $U\in [0,1]$ for ties.

1. Pool the observations into $z=(X_1,\dots ,X_m,Y_1,\dots ,Y_n)$ and compute $T(z)$.
2. Enumerate all $N!$ coordinate permutations $\pi$ of $N=m+n$. For each permutation, treat its first $m$ observations as the first group and its last $n$ as the second. Recompute the U-statistic and both variance estimates from those groups, apply the same zero-denominator rule, and compute $T(z^{\pi })$.
3. Return the inclusive p-value $p_{\mathrm{i}\mathrm{n}\mathrm{c}}=N{!}^{-1}\mathrm{\#}\{\pi :T(z^{\pi })\ge T(z)\}$ and reject when $p_{\mathrm{i}\mathrm{n}\mathrm{c}}\le \alpha$. If tie randomization was requested, instead return $p_U=N{!}^{-1}\{\mathrm{\#}\{\pi :T(z^{\pi })>T(z)\}+U\mathrm{\#}\{\pi :T(z^{\pi })=T(z)\}\}$ and reject when $p_U\le \alpha$.

This is complete finite enumeration, with permutations counted with their multiplicities even when observations repeat. It does not define a Monte Carlo approximation using a subset of permutations. No generic weak-null validity is asserted by the procedure alone; that requires the applicable studentization and estimator-consistency theorem.