  The problem of accurate nonparametric estimation of distributional
functionals (integral functionals of one or more probability distributions) has
received recent interest due to their wide applicability in signal processing,
information theory, machine learning, and statistics. In particular,
$k$-nearest neighbor (nn) based methods have received a lot of attention due to
their adaptive nature and their relatively low computational complexity. We
derive the mean squared error (MSE) convergence rates of leave-one-out $k$-nn
plug-in density estimators of a large class of distributional functionals
without boundary correction. We then apply the theory of optimally weighted
ensemble estimation to obtain weighted ensemble estimators that achieve the
parametric MSE rate under assumptions that are competitive with the state of
the art. The asymptotic distributions of these estimators, which are unknown
for all other $k$-nn based distributional functional estimators, are also
presented which enables us to perform hypothesis testing.
