The doubly generalized exponential-geometric frailty distribution

We proposed the doubly generalized exponential-geometric frailty (DGEGF) distribution, a hierarchical lifetime model for settings with a decreasing hazard, a random number of failure-prone components, and shared latent heterogeneity. The construction combined a geometric $ k $-out-of-$ n $ failure rule, in which the system failed at the $ k $th component failure rather than at the first, with gamma frailty acting on exponential component lifetimes. This hierarchy implied that every member of the family has a strictly decreasing failure rate, so the model was intended for burn-in or early-failure reliability data and for heterogeneous survival cohorts where risk decayed over time. We derived closed-form expressions for the marginal density, distribution, and survival functions and showed that the model was identifiable for both fixed and unknown $ k $. A Monte Carlo study over several parameter regimes indicated that the baseline rate and geometric parameter were accurately estimated in moderate samples, whereas the frailty parameter can be highly variable in small samples, in line with known numerical-identifiability issues in multi-parameter lifetime models. In four benchmark applications, we compared DGEGF with exponential, exponential-geometric, and shared-frailty alternatives. The results showed that, under decreasing hazards, DGEGF offered a transparent way to encode redundancy and unobserved heterogeneity while remaining competitive in fit. We also indicated how the same hierarchical construction can be coupled with Weibull or log-logistic baselines to accommodate non-monotone hazards when needed.