Nearest neighbor estimates of entropy

Motivated by the problems in molecular sciences, we introduce new nonparametric estimators of entropy which are based on the kth nearest neighbor distances between the n sample points, where k (< n - 1) is a fixed positive integer. These provide competing estimators to an estimator proposed by Kozachenko and Leonenko (1987), which is based on the first nearest neighbor distances of the sample points. These estimators are helpful in the evaluation of entropies of random vectors. We establish the asymptotic unbiasedness and consistency of the proposed estimators. For some standard distributions, we also investigate their performance for finite sample sizes using Monte Carlo simulations. The proposed estimators are applied to estimate the entropy of internal rotation in the methanol molecule, which can be characterized by a one-dimensional random vector, and of diethyl ether, which is described by a four-dimensional random vector.