  A problem of improving the accuracy of nonparametric entropy estimation for a
stationary ergodic process is considered. New weak metrics are introduced and
relations between metrics, measures, and entropy are discussed. Based on weak
metrics, a new nearest-neighbor entropy estimator is constructed and has a
parameter with which the estimator is optimized to reduce its bias. It is shown
that estimator's variance is upper-bounded by a nearly optimal Cramer-Rao lower
bound.
