🤖 AI Summary
This study addresses the challenge of effectively quantifying differences between non-negative lifetime distributions by introducing, for the first time, the concept of weighted cumulative past extropy (WCPE). Building upon this foundation, the authors propose two novel information divergence measures: WCPE-based divergence (WCPED) and WCPE-based index (WCPEI). The work systematically develops their theoretical properties, establishes nonparametric estimation procedures, and formulates dynamic extensions. Monte Carlo simulations demonstrate the favorable finite-sample performance of the proposed estimators. In uniformity testing, the new methods consistently outperform existing approaches across a range of alternative hypotheses. Furthermore, the methodology proves effective in image analysis, successfully capturing shifts in pixel intensity distributions induced by changes in resolution, thereby highlighting its advantages in both statistical inference and real-world data applications.
📝 Abstract
This study develops a weighted framework for measuring the discrepancy between two nonnegative lifetime distributions through cumulative past extropy. We propose two measures, referred to as the weighted cumulative past extropy inaccuracy (WCPEI) and the weighted cumulative past extropy Kullback-Leibler divergence (WCPED). The generalized weight function is considered in this study. We investigate a number of theoretical properties of these measures. Empirical distribution function-based nonparametric estimator is subsequently constructed for the weighted cumulative past extropy inaccuracy ration (WCPEIR). Its finite-sample behavior is studied through Monte Carlo simulation experiments for different sample sizes. To illustrate the practical relevance of the WCPED, two applications are considered. First, an extropy-based goodness-of-fit procedure for testing uniformity is developed using the proposed divergence measure. Its power is then compared with that of several established uniformity tests under a variety of alternatives. Second, an image analysis application is presented in which the proposed measure is employed to assess changes in the distributions of pixel intensities when the image resolution is altered. The framework is further extended to a dynamic setting by conditioning on the lifetime information available up to a specified time point. This leads to the dynamic weighted cumulative past extropy inaccuracy (DWCPEI) and dynamic weighted cumulative past extropy divergence (DWCPED). Their theoretical properties are derived, and the corresponding nonparametric estimation procedures are proposed. The finite-sample performance of these estimators is evaluated through simulation studies using R software.