Entropy-Based Evidence Functions for Testing Dilation Order via Cumulative Entropies

This paper introduces novel entropy-based evidence functions for assessing the dilation order of probability distributions, constructed from cumulative residual entropy (CRE) and cumulative entropy (CE). The proposed test statistics are developed within a rigorous evidential framework, and their asymptotic distributions are established, ensuring a solid foundation for large-sample inference. Beyond their theoretical appeal, these tests act as effective entropy-driven evidence functions, offering a compelling alternative to traditional approaches such as Kullback–Leibler discrepancies. Comprehensive Monte Carlo simulations highlight their robustness and consistently superior power across a wide range of distributional scenarios, including heavy-tailed models where conventional methods often fail. A real data example further illustrates their practical utility, showing how cumulative entropies can provide sharper statistical evidence and clarify stochastic comparisons in applied settings. These results not only advance the theoretical foundation of statistical evidence but also open avenues for applying cumulative entropies to broader classes of stochastic inference problems.