A Scale-Invariant Adaptive Test for IFR Alternatives Based on Cumulative Residual Entropy Under Proportional Hazards

Testing exponentiality against increasing failure rate alternatives is central to reliability theory and lifetime data analysis. This paper develops the Adaptive Cumulative Residual Entropy Test under Proportional Hazards (Adaptive CRE-PH Test), a scale-invariant nonparametric procedure that unifies a Tsallis-entropy departure functional with a proportional-hazards transformation. For each tuning value, the fixed-q statistic admits a normalized-spacing representation. Under exponentiality, the spacing proportions follow a Dirichlet distribution, yielding exact finite-sample means, covariances, and a joint null characterization of the adaptive maximum. Joint asymptotic normality and consistency under fixed alternatives are established for the complete dependent score vector. Structural analysis of the score family motivates a prespecified moderate grid governed by endpoint stability, directional diversity, and multiplicity economy, rather than retrospective power optimization. Monte Carlo experiments demonstrate accurate size control, explicitly quantify calibration stability, and show power close to the best fixed-q component under linear failure rate, Makeham, and Weibull alternatives. A dedicated power experiment confirms substantial detection capability against an IFRA-but-not-IFR benchmark, showing that the broader population sign condition has practical as well as theoretical relevance. Additional DFR and bathtub experiments clarify directional specificity: rejection provides evidence against exponentiality in the IFR-sensitive direction but does not, without shape-specific inference, establish a globally increasing hazard. Three real-data applications illustrate the practical importance of distinguishing formal directional inference from exploratory Q–Q and total-time-on-test diagnostics.