  Estimators of information theoretic measures such as entropy and mutual
information are a basic workhorse for many downstream applications in modern
data science. State of the art approaches have been either geometric (nearest
neighbor (NN) based) or kernel based (with a globally chosen bandwidth). In
this paper, we combine both these approaches to design new estimators of
entropy and mutual information that outperform state of the art methods. Our
estimator uses local bandwidth choices of $k$-NN distances with a finite $k$,
independent of the sample size. Such a local and data dependent choice improves
performance in practice, but the bandwidth is vanishing at a fast rate, leading
to a non-vanishing bias. We show that the asymptotic bias of the proposed
estimator is universal; it is independent of the underlying distribution.
Hence, it can be pre-computed and subtracted from the estimate. As a byproduct,
we obtain a unified way of obtaining both kernel and NN estimators. The
corresponding theoretical contribution relating the asymptotic geometry of
nearest neighbors to order statistics is of independent mathematical interest.
