estimators when $\mathbf{X}$ and $\mathbf{Y}$ may have a mixture of discrete and continuous components (cases 2 and 3) in Section \[sec:mixed\]. Second, in conjunction with Theorem \[thm:variance\] which follows, the results in Theorem \[thm:bias\] can be used to derive MI ensemble estimators in Appendix \[subsec:cont\_ensemble\] that achieve the parametric MSE convergence rate when the densities are sufficiently smooth. The expression in (\[eq:bias2\]) enables us to achieve the parametric rate under less restrictive smoothness assumptions on the densities ($s>d/2$ for (\[eq:bias2\]) compared to $s\geq d$ for (\[eq:bias1\])). The extra condition required on the mixed derivatives of $g$ to obtain the expression in (\[eq:bias2\]) is satisfied, for example, for Shannon and Renyi information measures.

\[thm:variance\](Variance) If the functional $g$ is Lipschitz continuous in both of its arguments with Lipschitz constant $C_{g}$, then the variance of $\gt$ is $$\var\left[\gt\right]\leq\frac{22C_{g}^{2}||K_{X}\cdot K_{Y}||_{\infty}^{2}}{N}.$$

Similar to Theorem \[thm:bias\], Theorem \[thm:variance\] is used to derive variance expressions for the MI plug-in estimators under cases 2 and 3. Theorem \[thm:variance\] is also necessary to derive optimally weighted ensemble estimators. The proofs of Theorems \[thm:bias\] and \[thm:variance\] are similar to the proofs of the bias and variance results for the divergence functional estimators in [@moon2016arxiv]. The primary difference is in 