The definition of kernels in nonparametric can be formulated as follows.
[Randles&Wolfe] pp.61-62. A parameter $\gamma$ is said to be estimable of degree $r$ for the family of distributions $\mathcal{F}$ if $r$ is the smallest sample size for which there exists a function $h^{*}(x_1,\cdots,x_r)$ such that the $\gamma$ can be unbiased estimated over the family. i.e. $$E_{F}[h^{*}(X_1,\cdots,X_r )]=\gamma,\forall F\in \mathcal{F}$$. Where $X_1,\cdots,X_r$ is a random sample from $F\in\mathcal{F}$ and $h^{*}$ is a statistics which does NOT depend on particular kernel $F$. WLOG, we can assume $h^{*}$ to be symmetric in its variables.
My question is that is, there is an obvious reason for preference of a kernel $h^{*}$ of smaller degree: And as observed, usually a higher degree kernel will lead to the same $U$-statistics $$U(X_1,\cdots,X_n):=\frac{1}{\left(\begin{array}{c} n\\ r \end{array}\right)}\sum_{\beta\in Perm(r)}h^{*}(X_{\beta(1)},\cdots,X_{\beta(n)})$$ as the minimal degree kernel $h^{**}$.
However, is there an example where the kernel $h^{**}$ of minimal degree and a higher degree kernel $h^{***}$ lead to different $U$-statistics? And if so, which one is prefereable and why?
References
[Randles&Wolfe]Randles, Ronald H., and Douglas A. Wolfe. Introduction to the theory of nonparametric statistics. Vol. 1. New York: Wiley, 1979.
