On Asymptotic Normality of Entropy Measures
Atıf Evren, Erhan Ustaoğlu

Abstract
Since distributions of qualitative variables can be represented by multinomial distributions, the role of multinomial distribution in entropy considerations is essential in statistics. Moreover for larger sample sizes multinomial distributions can be approximated well by multivariate normal distributions. The measures of qualitative variations depend on either class frequencies or some functional forms of class frequencies. Therefore the connection between qualitative variation statistics and normality seems straightforward for larger sample sizes. Asymptotic distributions of Shannon, Rényi and Tsallis entropies make some hypothesis testing and inferential techniques applicable to qualitative variations because some entropy measures are also frequently used in qualitative variation calculations. In this study, first we will give few examples of such applications by three entropy measures. Then we make a comparison between the performances of these three entropy measures. Finally, the degree of uncertainty, which is a significant factor that affects the speed of convergence to normality, is emphasized.

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