The variance of a U-statistic $\widehat{\Theta}$ (with fixed kernel) amounts to $Var(\widehat{\Theta}) = \sum_{c=1}^m \alpha_c \kappa_c - (1 - \alpha_0)\Theta^2$, where all parameters are defined as in Fuchs et al. (2020) and $\kappa_c$ and $\Theta^2$ can be estimated via U-statistics as well. Thus, by plugging these U-statistics into the formula for $Var(\widehat{\Theta})$ we can estimate the variance of $\widehat{\Theta}$.
U-statistics are defined as having fixed, non-random kernels. So my question would be, whether the above results might also hold for a kernel that depends on a random vector $w$?
Reference: M. Fuchs, R. Hornung, A.-L. Boulesteix, R. De Bin (2020). On the asymptotic behaviour of the variance estimator of a U-statistic. Journal of Statistical Planning and Inference, vol. 209, 101-111
