handling certain products of the marginal KDEs that appear in the expansion of the MSE. See Appendix \[sec:biasProof\] and \[sec:VarProof\] for details.

Theorems \[thm:bias\] and \[thm:variance\] indicate that for the MSE of the plug-in estimator to go to zero for case 1, we require $h_{X},h_{Y}\rightarrow0$ and $Nh_{X}^{d_{X}}h_{Y}^{d_{Y}}\rightarrow\infty$. The Lipschitz assumption on $g$ is comparable to other nonparametric estimators of distributional functionals [@kandasamy2015nonparametric; @singh2014exponential; @singh2014renyi; @moon2016arxiv; @krishnamurthy2014divergence]. Specifically, assumption $\mathcal{A}.1$ ensures that functionals such as those for Shannon and Renyi informations are Lipschitz on the space $\epsilon_{0}$ to $\epsilon_{\infty}$.

Mixed Random Variables {#sec:mixed}
======================

In this section, we extend the results of Section \[sec:MI\_est\] to MI estimation when $\mathbf{X}$ and $\mathbf{Y}$ may have a mixture of discrete and continuous components. For simplicity, we focus primarily on the important case when $\mathbf{X}$ is continuous and $\mathbf{Y}$ is discrete (case 2 in Section \[sec:intro\]). The more general case when $\X$ and $\Y$ may have any mixture of continuous and discrete components (case 3 in Section \[sec:intro\]) is discussed in Section \[subsec:general\_rates\]. As an example of the former case, if $\mathbf{Y}$ is a predictor variable (e.g. classification labels), then the MI between $\mathbf{X}$ and $\mathbf{Y}$ indicates the value of $\mathbf{X}$ as a predictor of 