Preservation of relative aging orders under Cumulative-Hazard transformations

Preservation under a cumulative-hazard transformation is generally a sufficient-condition problem; universal preservation asks which generators work for every admissible baseline pair. We characterized this stronger property for the aging-faster (AF) and aging-faster-in-average (AFA) orders. For a proper survival generator $ {\mathit{\Phi}} $, put $ {\mathit{\Omega}} = -\mathrm{l}\mathrm{o}\mathrm{g}{\mathit{\Phi}} $, $ \omega = {\mathit{\Omega}} {'} $, $ {\eta }_{\omega }\left(u\right) = u\omega {'}\left(u\right)/\omega \left(u\right) $, and $ {\kappa }_{{\mathit{\Omega}} }\left(u\right) = u\omega \left(u\right)/{\mathit{\Omega}} \left(u\right) $. Among proper baselines with positive interior hazards and $ X{\ge }_{\mathrm{s}\mathrm{t}}Y $, universal AF preservation held if and only if $ \omega $ was nondecreasing and $ {\eta }_{\omega } $ was nonincreasing; universal AFA preservation held if and only if $ {\kappa }_{{\mathit{\Omega}} } $ was nonincreasing. The sufficient directions were established distortion-preservation results. Our constructive necessity argument used proportional exponential baselines and smooth hazard ratios reaching one. Different-generator comparisons were followed by transitivity, and gamma thresholds recovered known AF and AFA upper-record comparisons in nonhomogeneous Poisson epoch-time form. Analytical power-threshold consequences and a family-specific eventual-failure result distinguished global conclusions from finite-horizon numerical observations. A reproducible censored component-life-test calibration illustrated model-based use of the criteria while separating fitted-model implications from evidence about the underlying populations.