Measures of qualitative variation in the case of maximum entropy

Abstract

Asymptotic behavior of qualitative variation statistics, including entropy measures, can be modeled well by normal distributions. In this study, we test the normality of various qualitative variation measures in general. We find that almost all indices tend to normality as the sample size increases, and they are highly correlated. However, for all of these qualitative variation statistics, maximum uncertainty is a serious factor that prevents normality. Among these, we study the properties of two qualitative variation statistics; VarNC and StDev statistics in the case of maximum uncertainty, since these two statistics show lower sampling variability and utilize all sample information. We derive probability distribution functions of these statistics and prove that they are consistent. We also discuss the relationship between VarNC and the normalized form of Tsallis (α = 2) entropy in the case of maximum uncertainty.