Nonparametric von Mises Estimators for Entropies, Divergences and Mutual Informations

We propose and analyse estimators for statistical functionals of one or more distributions under nonparametric assumptions. Our estimators are derived from the von Mises expansion and are based on the theory of influence functions, which appear in the semiparametric statistics literature. We show that estimators based either on data-splitting or a leave-one-out technique enjoy fast rates of convergence and other favorable theoretical properties. We apply this framework to derive estimators for several popular information theoretic quantities, and via empirical evaluation, show the advantage of this approach over existing estimators.