Nonparametric Tests for Exponentiality Against DMRL Alternatives Based on Generalized Cumulative Residual Tsallis Entropy

A family of nonparametric tests is developed for testing exponentiality against decreasing mean residual life (DMRL) alternatives using generalized cumulative residual Tsallis entropy (GCRTE). For entropy-order pairs in a theoretically certified region, a sharp population inequality guarantees that the associated departure functional is non-negative over the DMRL class and vanishes if and only if the lifetime distribution is exponential. The exact empirical plug-in admits a weighted order-statistic representation, while the interior-point smooth-score version used in the numerical work is first-order asymptotically equivalent under the stated regularity conditions. Normalization by the sample mean yields exact scale invariance and removes dependence on the unknown exponential scale parameter. For bounded sufficiently regular scores, first-order asymptotic normality is established, and consistency holds against fixed nonexponential DMRL alternatives within the certified parameter region. A Kaplan–Meier extension with explicit tail completion is developed for independently right-censored data, with exact conditional null calibration under exponential censoring. Monte Carlo experiments examine both theoretically certified and exploratory entropy-order configurations under linear failure rate, gamma, Weibull, and additional DMRL alternatives using finite-sample null calibration, with empirical size checks and censoring fractions from 10% to 40%. Two complete-data applications illustrate the procedure: exponentiality is not rejected for the epidemic-infection dataset, whereas the leukemia survival-time data show a significant departure in the DMRL direction. Overall, the framework provides a flexible scale-invariant approach whose directional DMRL interpretation is explicitly separated from the regularity conditions required for large-sample inference.