or $K_{Y}$, $\mathcal{S}$ either $\mathcal{S}_{X}$ or $\mathcal{S}_{Y}$, $h$ either $h_{X}$ or $h_{Y}$. Let $p_{x}(u):\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a polynomial in $u$ of order $q\leq r=\left\lfloor s\right\rfloor $ whose coefficients are a function of $x$ and are $r-q$ times differentiable. For any positive integer $t$ $$\int_{x\in\mathcal{S}}\left(\int_{u:K(u)>0,\,x+uh\notin\mathcal{S}}K(u)p_{x}(u)du\right)^{t}dx=v_{t}(h),$$ where $v_{t}(h)$ admits the expansion $$v_{t}(h)=\sum_{i=1}^{r-q}e_{i,q,t}h^{i}+o\left(h^{r-q}\right),$$ for some constants $e_{i,q,t}$.

Assumption $(\mathcal{A}.5)$ states that the support of the density is smooth with respect to the kernel $K$ in the sense that the expectation with respect to any random variable $u$ of the area of the kernel that falls outside the support $\mathcal{S}$ is a smooth function of the bandwidth $h$ provided that the distribution function $p_{x}(u)$ of $u$ is smooth (e.g. $s\geq2$). The inner integral captures this expectation while the outer integral averages this inner integral over all points near the boundary of the support. The $v_{t}(h)$ term captures the fact that the smoothness of this expectation is proportional to the smoothness of the function $p_{x}(u)$. As an example, this smoothness assumption is satisfied when the support is rectangular and the kernel is the uniform rectangular kernel [@moon2016arxiv; @moon2016isit]. Note that this boundary assumption does not result in parametric convergence rates for the plug-in estimator $\gt$, which 