Modeling of bounded data via order statistics: extropy, Fisher information, simulation, and applications

Abstract

Bounded data on the unit interval, arising from proportions, rates, and normalized lifetimes, are ubiquitous in reliability engineering, biomedical research, and applied sciences, yet the information-theoretic tools needed to fully exploit flexible unit-interval models remain underdeveloped. This paper presents a unified information-theoretic treatment of the unit extended exponential distribution (UEExpD), a versatile bounded lifetime model, built around extropy and its weighted, cumulative residual, and negative cumulative variants. We derive closed-form expressions for these measures for the UEExpD itself and extend them to its order statistics through one-dimensional integral representations, evaluated by numerical quadrature, which characterize the information content across ordered and extreme observations. The Fisher information matrix is obtained and examined as a function of sample order, identifying which observations are most informative for parameter estimation. Maximum likelihood estimation is used throughout, with extensive simulation studies assessing bias, mean square error, and relative error across sample sizes and supporting the consistency of the estimators; every closed-form result is independently validated against numerical benchmarks. The framework is further illustrated on two real datasets, to which the UEExpD provides an adequate fit, demonstrating that the UEExpD offers a theoretically rich and practically useful tool for information-theoretic and reliability analysis of bounded data.