October 2015, Vol. 27, No. 10, Pages 2097-2106
We compare an entropy estimator Ĥz recently discussed by Zhang (2012) with two estimators, Ĥ1 and Ĥ2, introduced by Grassberger (2003) and Schürmann (2004). We prove the identity Ĥz ≡ Ĥ1, which has not been taken into account by Zhang (2012). Then we prove that the systematic error (bias) of Ĥ1 is less than or equal to the bias of the ordinary likelihood (or plug-in) estimator of entropy. Finally, by numerical simulation, we verify that for the most interesting regime of small sample estimation and large event spaces, the estimator Ĥ2 has a significantly smaller statistical error than Ĥz.