Studentized two-sample U-statistic and permutation distribution

For the two-sample statistic in Two-sample order-r U-statistic and first projections, let ${\widehat{\sigma }}_m^2$ and ${\widehat{\sigma }}_n^2$ be nonnegative measurable statistics of the two samples intended to estimate the variances of $g_X(X_1)$ and $g_Y(Y_1)$. Each may depend on both groups; separate-group dependence is an additional property, not part of this definition. Define the estimated standard error and studentized statistic by

$V_{m,n}^{\mathrm{S}\mathrm{E}}:=r\sqrt{\frac{{\widehat{\sigma }}_m^2}{m}+\frac{{\widehat{\sigma }}_n^2}{n}},S_{m,n}:=\{\begin{array}{ll}{U}_{m,n}\mathrm{/}{V}_{m,n}^{\mathrm{S}\mathrm{E}},& V_{m,n}^{\mathrm{S}\mathrm{E}}>0,\\ 0,& V_{m,n}^{\mathrm{S}\mathrm{E}}=0.\end{array}$

For a uniform random permutation $\pi$ of the pooled $N=m+n$ observations, recompute both group statistics and both variance estimates on the permuted groups; the permutation distribution of $S_{m,n}$ is the conditional law of $S_{m,n}^{\pi }$ given the pooled observations.

The report's scale is $V_{m,n}^{\mathrm{C}\mathrm{R}}=r\sqrt{{\widehat{\sigma }}_m^2+(m\mathrm{/}n){\widehat{\sigma }}_n^2}$ and its statistic is $U_{m,n}\mathrm{/}{V}_{m,n}^{\mathrm{C}\mathrm{R}}$, with a $\sqrt{m}$ multiplier. Since $V_{m,n}^{\mathrm{S}\mathrm{E}}=V_{m,n}^{\mathrm{C}\mathrm{R}}\mathrm{/}\sqrt{m},$ the usual standard-error studentization is $S_{m,n}=\sqrt{m}{U}_{m,n}\mathrm{/}{V}_{m,n}^{\mathrm{C}\mathrm{R}}\mathrm{.}$ The standard-normal limits for studentized statistics use $S_{m,n}$ itself as the pivot, under the null, nondegeneracy, and variance-consistency hypotheses of the corresponding limit theorem. An additional $\sqrt{m}$ multiplying this $S_{m,n}$ is not part of the studentized pivot.

Consistency and permutation consistency are hypotheses of asymptotic validity results, not consequences of this definition. For an estimate depending on both groups, consistency on an IID pooled-mixture row alone does not imply permutation consistency; that condition must be proved for the actual estimate.