is in contrast with the boundary assumptions in [@singh2014exponential; @singh2014renyi; @krishnamurthy2014divergence; @kandasamy2015nonparametric]. However, the estimators in [@singh2014exponential; @singh2014renyi; @krishnamurthy2014divergence; @kandasamy2015nonparametric] perform boundary correction, which requires knowledge of the density support boundary and complex calculations at the boundary in addition to the boundary assumptions, to achieve the parametric convergence rates. In contrast, we use ensemble methods to improve the resulting convergence rates of $\gt$ without boundary correction.

\[thm:bias\](Bias) Under assumptions $\mathcal{A}.0-\mathcal{A}.5$ and for general $g$, the bias of $\gt$ is $$\begin{aligned}
\bias\left[\gt\right] & = & \sum_{\substack{j=0\\
i+j\neq0
}
}^{r}\sum_{i=0}^{r}c_{10,i,j}h_{X}^{i}h_{Y}^{j}+\frac{c_{11}}{Nh_{X}^{d_{X}}h_{Y}^{d_{Y}}}\nonumber \\
 &  & +O\left(h_{X}^{s}+h_{Y}^{s}+\frac{1}{Nh_{X}^{d_{X}}h_{Y}^{d_{Y}}}\right).\label{eq:bias1}\end{aligned}$$ If $g\left(t_{1},t_{2}\right)$ also has $j,l$-th order mixed derivatives $\frac{\partial^{j+l}}{\partial t_{1}^{j}\partial t_{2}^{l}}$ that depend on $t_{1}$ and $t_{2}$ only through $t_{1}^{\alpha}t_{2}^{\beta}$ for some $\alpha,\beta\in\mathbb{R}$ for each $1\leq k,l,\leq\lambda$, the bias of $\gt$ is $$\begin{aligned}
 &  & \lefteqn{\bias\left[\gt\right]}\nonumber \\
 &  & =\sum_{\substack{m,n=0\\
i+j+m+n\neq0
}
}^{\left\lfloor \lambda/2\right\rfloor }\sum_{i,j=0}^{r}c_{11,j,i,m,n}\frac{h_{X}^{i}h_{Y}^{j}}{\left(Nh_{X}^{d_{X}}\right)^{m}\left(Nh_{Y}^{d_{Y}}\right)^{n}}\nonumber \\
 &  & +\sum_{m=1}^{\left\lfloor \lambda/2\right\rfloor }\sum_{i=0}^{r}\sum_{j=0}^{r}c_{13,m,n,j}h_{X}^{i}h_{Y}^{j}/\left(Nh_{X}^{d_{X}}h_{Y}^{d_{Y}}\right)^{m}\nonumber \\
 &  & +O\left(h_{X}^{s}+h_{Y}^{s}+1/\left(Nh_{X}^{d_{X}}h_{Y}^{d_{Y}}\right)^{\lambda/2}\right).\label{eq:bias2}\end{aligned}$$

The constants in both (\[eq:bias1\]) and (\[eq:bias2\]) depend only on the densities and their derivatives, the functional $g$ and its derivatives, and the kernels. They are independent of $N,$ $h_{X}$, and $h_{Y}.$

The purpose of Theorem \[thm:bias\] is two-fold. First, we use Theorem \[thm:bias\] to derive the bias expressions for the MI plug-in 