Symmetrization of a U-statistic kernel
= Symmetrization of a U-statistic kernel
{title2=$h_s(x_1,\ldots,x_r)=\frac1{r!}\sum_\pi h(x_{\pi(1)},\ldots,x_{\pi(r)})$}
Exchangeability of an independent identically distributed sample makes every permuted kernel have the same expectation. Averaging over permutations therefore leaves the target functional unchanged and gives a symmetric kernel. The <U-statistic> then averages this kernel over the unordered r-element sample subsets.