of $30\leq L\leq60$, and setting $\mathcal{L}$ to be $L$ linearly spaced values between the minimum and maximum values described above works well in practice.

The resulting ensemble estimators are robust in the sense that they are not sensitive to the exact choice of the bandwidths or the number of estimators as long as the the rough rules-of-thumb given above are followed. Moon et al [@moon2016arxiv; @moon2016isit] gives more details on ensemble estimator parameter selection for continuous divergence estimation. These details also apply to the continuous parts of the mixed cases for MI estimation in this paper.

Since the optimal weight $w_{0}$ can be calculated offline, the computational complexity of the estimators is dominated by the construction of the KDEs which has a complexity of $O\left(N^{2}\right)$ using the standard implementation. For very large datasets, more efficient KDE implementations (e.g. [@raykar2010fast]) can be used to reduce the computational burden.

Central Limit Theorem {#subsec:clt}
---------------------

We finish this section with central limit theorems for the ensemble estimators. This enables us to perform hypothesis testing on the mutual information.

\[thm:clt\] Let $\g w^{cont}$ be a weighted ensemble estimator when $\X$ and $\Y$ are continuous with bandwidths $h_{X}(l_{X})$ and $h_{Y}(l_{Y})$ for each estimator in the 