Improving convergence of divergence functional ensemble estimators

Recent work has focused on the problem of non-parametric estimation of divergence functionals. Many existing approaches are restrictive in their assumptions on the density support or require difficult calculations at the support boundary which must be known a priori. We derive the MSE convergence rate of a leave-one-out kernel density plug-in divergence functional estimator for general bounded density support sets where knowledge of the support boundary is not required. We generalize the theory of optimally weighted ensemble estimation to derive two estimators that achieve the parametric rate when the densities are sufficiently smooth. The asymptotic distribution of these estimators and tuning parameter selection guidelines are provided. Based on the theory, we propose an empirical estimator of Rényi-α divergence that outperforms the standard kernel density plug-in estimator, especially in higher dimensions.