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

This paper introduces novel non-parametric entropy-based evidence functions and associated test statistics for assessing the dilation order of probability distributions constructed from cumulative residual entropy and cumulative entropy. The proposed evidence functions are explicitly tuned to questions about distributional variability and stochastic ordering, rather than global model fit, and are developed within a rigorous evidential framework. Their asymptotic distributions are established, providing a solid foundation for large-sample inference. Beyond their theoretical appeal, these procedures act as effective entropy-driven tools for quantifying statistical evidence, offering a compelling non-parametric alternative to traditional approaches, such as Kullback–Leibler discrepancies. Comprehensive Monte Carlo simulations highlight their robustness and consistently high power across a wide range of distributional scenarios, including heavy-tailed models, where conventional methods often perform poorly. 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. Altogether, these results advance the theoretical foundation of evidential statistics and open avenues for applying cumulative entropies to broader classes of stochastic inference problems.