Simple SummaryWe obtain expressions for the asymptotic distributions of the Rényi and Tsallis of orderqentropies, and Fisher information when computed on the maximum likelihood estimator of probabilities from multinomial random samples. We recall results related to the Shannon entropy. We build a test for comparing entropies of different types and categories.AbstractWe obtain expressions for the asymptotic distributions of the Rényi and Tsallis of orderqentropies and Fisher information when computed on the maximum likelihood estimator of probabilities from multinomial random samples. We verify that these asymptotic models, two of which (Tsallis and Fisher) are normal, describe well a variety of simulated data. In addition, we obtain test statistics for comparing (possibly different types of) entropies from two samples without requiring the same number of categories. Finally, we apply these tests to social survey data and verify that the results are consistent but more general than those obtained with aχ2test.