Empirical Squared Hellinger Distance Estimator and Generalizations to a Family of α-Divergence Estimators

We present an empirical estimator for the squared Hellinger distance between two continuous distributions, which almost surely converges. We show that the divergence estimation problem can be solved directly using the empirical CDF and does not need the intermediate step of estimating the densities. We illustrate the proposed estimator on several one-dimensional probability distributions. Finally, we extend the estimator to a family of estimators for the family of $\alpha$-divergences, which almost surely converge as well, and discuss the uniqueness of this result. We demonstrate applications of the proposed Hellinger affinity estimators to approximately bounding the Neyman–Pearson regions.