Influence function of a U-statistic ($\mathrm{IF}(x;T,F)=m\{\mathbb{E}[h(x,X_2,\dots ,X_m)]-T(F)\}$)

For a symmetric degree-$m$ U-statistic kernel, the population functional is $T(F)=\int hd{F}^m$. Contaminating each of the $m$ identical distribution arguments and differentiating gives the displayed influence function. It equals $m$ times the first Hoeffding projection and gives leading asymptotic variance $m^2\mathrm{Var}(h_1(X))/n$. If that projection vanishes, the usual root-$n$ normal limit degenerates and higher-order projections determine the limiting law.