This paper introduces a new flexible lifetime distribution that unifies the exponential-geometric and Lomax models through a random system size and a shared gamma frailty. It models the first failure time in a system with a geometrically distributed number of conditionally exponential components with gamma frailty, producing a geometric mixture of Lomax distributions. The model generalizes classical distributions, reducing to exponential-geometric, Lomax, or exponential under appropriate limits. We derived analytical expressions for the cumulative distribution, survival, and hazard functions, discussed maximum likelihood estimation, and illustrated its competitive and versatile fits on real datasets.

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.

We propose a new lifetime distribution for first-failure times in heterogeneous systems by combining exponential component lifetimes with a shared gamma frailty and a logarithmic distribution governing the latent number of competing failure causes. The model is constructed as the minimum lifetime among a random number of conditionally exponential components, where the system size follows a logarithmic distribution, and all components share an unobserved gamma-distributed risk factor. The resulting marginal distribution is a logarithmic mixture of Lomax distributions and admits tractable series expressions for the probability density, distribution, survival, and hazard functions. We show that the conditional model $Y\mid N=n$ has a strictly decreasing failure rate and exhibits heavy-tailed behavior. We investigate the marginal hazard behavior of the proposed mixture numerically, and classical models such as the exponential–logarithmic, Lomax, and exponential distributions arise as limiting cases. Maximum likelihood and expectation–maximization (EM)-type estimation procedures are developed, and the flexibility of the model is illustrated through simulation and a real data application. In first-failure-only settings, the logarithmic mixing parameter $\eta$ may be weakly identifiable and therefore difficult to estimate reliably from the observed data alone.