The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

A new semiparameteric model of the receiver operating characteristic (ROC) curve based on the resilience family or proportional reversed hazard family is proposed, which is an alternative to the existing models. The resulting ROC curve and its summary indices (such as area under the curve and Youden index) have simple analytic forms. The partial likelihood method is applied to estimate the ROC curve. Moreover, the estimation methodologies of the resilience family of the ROC curve have been developed based on area under the curve estimators exploiting Mann–Whitney statistics and the Rojo approach. A simulation study has been carried out to assess the performance of all considered estimators. Real data from the American National Health and Nutrition Examination Survey has been analyzed in detail based on the proposed model and the usual binormal model prevalent in the literature. Real data in the context of brain injury-related biomarkers are also analyzed in order to compare our model with the Lehmann family of the ROC curves. Finally, we show that the proposed model may be applicable in the misspecification scenario through the Ducheme muscular dystrophy data.