Efficiency of entropy testing

Recently it was shown that Shannon entropy is more Bahadur efficient than any Renyi entropy of order alpha > 1. In this paper we shall show that relative Bahadur efficiency between any two Renyi entropies of orders alpha isin [0; 1] is 1 when the relative Bahadur efficiency is defined according to [1]. Despite the fact that the relative Bahadur efficiency is 1 it is shown that in a certain sense Shannon entropy is more efficient than Renyi entropy for alpha isin [0; 1]. This indicates that the definition of relative efficiency given in [1] does not fully capture the notion of efficiency.