It is generally assumed that the Rényi entropy is maximized by the q-exponentials and is hence useful to construct a generalized statistical mechanics. However, to the best of our knowledge, this assumption has never been explicitly checked. In this work, we consider the Rényi entropy with the linear and escort mean value constraints and check whether it is indeed maximized by q-exponentials. We show, both theoretically and numerically, that the Rényi entropy yields erroneous inferences concerning the optimum distributions of the q-exponential form and moreover exhibits high estimation errors in the regime of long range correlations. Finally, we note that the Shannon entropy successfully detects the power law distributions when the logarithmic mean value constraint is used.
