Efficiency of Entropy Testing
Peter Harremoes and I. Vajda
Recently it was shown that Shannon entropy is more Bahadur efficient than any R\'{e}nyi entropy of order $\alpha>1.$ In this paper we shall show that relative Bahadur efficiency between any two R\'{e}nyi entropies of orders $\alpha\in\left] 0;1\right]$ is 1 when the relative Bahadur efficiency is defined according to \cite{QuiRob85}. Despite the fact that the relative Bahadur efficiency is 1 it is shown that in a certain sense Shannon entropy is more efficient than R\'{e}nyi entropy for $\alpha\in\left] 0;1\right[ .$ This indicates that the definition of relative efficiency given in \cite{QuiRob85} does not fully capture the notion of efficiency.