A logarithmic–gamma frailty model for first-failure times with heavy-tailed risk

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.