Impact of Generalized Order Statistics and Its Dual on Entropy Estimation for the Exponentiated Generalized Pham Distribution: Theory and Applications.

Entropy is a fundamental measure of uncertainty in reliability and lifetime analysis, and the structure of the observed data intrinsically influences its estimation. In many practical applications, inference relies on ordered or record-based samples, for which generalized order statistics and their dual form provide a comprehensive and unifying framework encompassing order statistics, reversed order statistics, and record values as special cases. Despite substantial progress in entropy estimation and lifetime modeling, little attention has been devoted to a unified entropy estimation framework for flexible lifetime models under generalized ordered sampling schemes. This paper investigates the impact of generalized order statistics and their dual form on the estimation of entropy measures for the exponentiated generalized Pham distribution. The proposed distribution extends the classical Pham model through additional shape flexibility, enabling it to accommodate diverse reliability behaviors and heterogeneous tail characteristics. Several fundamental properties are derived, and maximum likelihood estimation and corresponding confidence intervals for model parameters and entropy measures are developed under both generalized order statistics and dual generalized order statistics frameworks. The general results are further specialized to order statistics, reversed order statistics, and upper and lower record values. Applications to two real datasets demonstrate that the proposed distribution provides an excellent fit compared with competing models, as confirmed by goodness-of-fit measures. The findings underscore the pivotal structural role of generalized order statistics and their dual form in entropy-based inference, particularly for record or partially observed data, where the sampling design critically affects uncertainty quantification.