Nonparametric Estimation of Extropy, R\'enyi Extropy, and Tsallis Extropy: Almost Sure Convergence and Asymptotic Normality

This paper proposes a nonparametric estimation procedure for extropy and its extensions, namely the {\alpha}-R'enyi and {\alpha}-Tsallis extropies, for finite discrete random variables. We establish almost sure rates of convergence and asymptotic normality for the plug-in estimators. The theoretical results are validated through a comprehensive simulation study. The findings provide a solid foundation for the use of extropy-based measures in practical applications, including forecasting, risk assessment, and decision-making under uncertainty.