A class of estimators of the Rényi and Tsallis entropies of an unknown distribution f in ℝm is presented. These estimators are based on the kth nearest-neighbor distances computed from a sample of N i.i.d. vectors with distribution f. We show that entropies of any order q, including Shannon’s entropy, can be estimated consistently with minimal assumptions on f. Moreover, we show that it is straightforward to extend the nearest-neighbor method to estimate the statistical distance between two distributions using one i.i.d. sample from each.
"A class of Rényi information estimators for multidimensional densities." Ann. Statist. 36 (5) 2153 - 2182, October 2008. https://doi.org/10.1214/07-AOS539