Exponential Concentration of a Density Functional Estimator

We analyze a plug-in estimator for a large class of integral functionals of one or more continuous probability densities. This class includes important families of entropy, divergence, mutual information, and their conditional versions. For densities on the d-dimensional unit cube [0,1]d that lie in a β-H"older smoothness class, we prove our estimator converges at the rate O left( n^-fracββ + d right). Furthermore, we prove the estimator is exponentially concentrated about its mean, whereas most previous related results have proven only expected error bounds on estimators.