This PhD report deals with the estimation of both Shannon entropy of distributions from independent and Markovian data and entropy rate of pure jump Markov processes with finite state space. In the latter case, different schemes of continuous and discrete observation of the processes are considered. Several related problems are studied. Kullback-Leibler information geometry linked with escort transformations come ahead of estimation. Others appear as applications of the estimation results. Tests on the entropy level of a distribution are derived from a large deviation principle satisfied by the sequence of empirical estimators of the entropy of a distribution. Properties related to entropy of birth and death Markovian queueing systems are also considered.
