plug-in MI estimator, respectively. It is shown in Section \[sec:mixed\_ensemble\] that the MSE of the plug-in estimator converges very slowly to zero under this setting. However, Theorems \[thm:bias\_mixed\] and \[thm:var\_mixed\] provide with us the necessary information for applying the theory of optimally weighted ensemble estimation to obtain estimators with improved rates. This is done in Section \[sec:mixed\_ensemble\].

Extension to Other Cases {#subsec:general_rates}
------------------------

The results in Section \[subsec:mixed\_conv\] can be extended to the case where $\mathbf{X}$ and/or $\mathbf{Y}$ may have a mixture of continuous and discrete components (case 3 in Section \[sec:intro\]). This scenario can be divided further into three different cases: A) $\X$ is continuous and $\Y$ has a mixture of discrete and continuous components; B) $\X$ and $\Y$ both have a mixture of discrete and continuous components; C) $\Y$ is discrete and $\X$ has a mixture of discrete and continuous components. Consider case A first. Denote the discrete and continuous components of $\mathbf{Y}$ as $\mathbf{Y}_{1}$ and $\mathbf{Y}_{2}$, respectively. Denote the respective support sets as $\mathcal{S}_{Y_{1}}$ and $\mathcal{S}_{Y_{2}}$. We can then write $$\begin{aligned}
 & \lefteqn{G_{3A}(\mathbf{X};\mathbf{Y})}\nonumber \\
 & =\sum_{y_{1}\in\mathcal{S}_{Y_{1}}}\int g\left(\frac{f_{X}(x)f_{Y}(y_{1},y_{2})}{f_{XY}(x,y_{1},y_{2})}\right)f_{XY}(x,y_{1},y_{2})dxdy_{2}\nonumber \\
 & =\sum_{y_{1}\in\mathcal{S}_{Y_{1}}}f_{Y_{1}}(y_{1})\int g\left(\frac{f_{X}(x)f_{Y_{2}|Y_{1}}(y_{2}|y_{1})}{f_{XY_{2}|Y_{1}}(x,y_{2}|y_{1})}\right)\nonumber \\
 & \times f_{XY_{2}|Y_{1}}(x,y_{2}|y_{1})dxdy_{2}.\label{eq:mixed_general}\end{aligned}$$ The subscript $3A$ indicates that we are considering case A under the 