Finite-Sample Bias Correction for Plug-in Estimators of Extropy, R\'enyi Extropy, and Tsallis Extropy

Abstract

This paper studies the finite-sample bias of plug-in estimators for Shannon, R\'enyi, and Tsallis extropies for finitely supported discrete random variables. We derive explicit first-order bias formulas for all three estimators by analyzing the bias of the intermediate functional S_\alpha = \sum_{i=1}^r (1-p_i)^\alpha. Bias-corrected estimators are then proposed. We show that the uncorrected and corrected estimators are asymptotically equivalent, differing only by terms of order O(1/n). A simulation study validates the theoretical results and demonstrates the superior finite-sample performance of the bias-corrected estimators. The results justify the use of the simpler uncorrected plug-in estimators in asymptotic settings, such as those considered in the companion paper on almost sure convergence and asymptotic normality.