Entropy is playing a fundamental role not only in information and communication theory, but also in many other fields like cryptography, neuroscience, and bioinformatics. An important task for many applications is to estimate the entropy of an unknown distributions given some samples.

The main topic of this thesis is the problem to estimate the entropy in the case of undersampling, that is, the number of samples is smaller as the support size of the distribution.

To solve this problem we settle on an approach that applies the James-Stein shrinkage principle to estimate a probability distribution with known support size. This approach is then generalized to the case where the support size is unknown. This generalization is achieved by using an appropriate support size estimator. Moreover, a new entropy estimator that applies the James-Stein shrinkage principle within a different perspective is developed. The performance of these two estimators is compared with other entropy estimators for different distributions and different sampling scenarios.