  Estimating expected polynomials of density functions from samples is a basic
problem with numerous applications in statistics and information theory.
Although kernel density estimators are widely used in practice for such
functional estimation problems, practitioners are left on their own to choose
an appropriate bandwidth for each application in hand. Further, kernel density
estimators suffer from boundary biases, which are prevalent in real world data
with lower dimensional structures. We propose using the fixed-k nearest
neighbor distances for the bandwidth, which adaptively adjusts to local
geometry. Further, we propose a novel estimator based on local likelihood
density estimators, that mitigates the boundary biases. Although such a choice
of fixed-k nearest neighbor distances to bandwidths results in inconsistent
estimators, we provide a simple debiasing scheme that precomputes the
asymptotic bias and divides off this term. With this novel correction, we show
consistency of this debiased estimator. We provide numerical experiments
suggesting that it improves upon competing state-of-the-art methods.
