Analysis of entropy and Fisher information for the new X-Lindley model with applications to failure data

Many disciplines, including environmental sciences, reliability engineering, hydrology, and information theory, rely heavily on information measures derived from statistical distributions for decision-making, risk assessment, and system design. Motivated by this need, the present study provides a comprehensive analysis of traditional and generalized information measures within the framework of the New X-Lindley distribution (NXLID). We systematically explore key concepts, including Fisher information, Shannon entropy, Tsallis entropy, Rényi entropy, extropy, and their cumulative and weighted counterparts, providing analytical expressions for each measure. Extensive simulation studies complement theoretical derivations to evaluate the performance of the proposed estimators in terms of bias, mean squared error, and mean relative error. To demonstrate the practical utility of the NXLID and its associated information measures, we analyze two real-world datasets related to component failure times in reliability contexts. In addition, we investigate the behavior of these information measures concerning order statistics, providing deeper insights into their distributional properties and variability. The findings underscore the flexibility and effectiveness of the NXLID in modeling lifetime data and assessing system reliability, establishing it as a valuable tool for both theoretical exploration and applied statistical modeling.