We propose and analyze estimators for statistical functionals of one or more
distributions under nonparametric assumptions. Our estimators 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.