A Frailty-Augmented Weibull–Poisson Competing Risk Model for \(k\)-out-of-\(n\) System Reliability: Properties, Bayesian Estimation, and Simulation

This paper introduces the Weibull–Poisson frailty (WPF) distribution, a four-parameter lifetime model in which a component fails at the earliest of the failure times of a zero-truncated Poisson number of latent causes, each cause having a Weibull failure time modulated by its own gamma frailty. The system lifetime is the \(k\)-th order statistic of \(n\) independent component lifetimes, so the model applies directly to \(k\)-out-of-\(n\) systems. The survival, density and hazard functions are available in closed form, and the model nests six distributions, including the Weibull–Poisson and gamma-frailty Weibull models. We characterize the possible hazard shapes (decreasing, increasing and unimodal), establish a stochastic ordering in the scale, competing-risk and frailty parameters, derive the condition under which the moments are finite, and show that ignoring the frailty while keeping the other parameters fixed always underestimates the mean time to failure (MTTF), most strongly for components with a decreasing hazard. Bayesian estimation is carried out with an adaptive Metropolis-within-Gibbs sampler. A Monte Carlo study shows that the Bayesian estimator recovers the shape parameter and the MTTF accurately and is far more stable than maximum likelihood, whereas the scale and competing-risk parameters are only weakly identified. Two applications illustrate the model: the frailty component is not supported by Aarset’s device-lifetime data, but it significantly improves the fit to Badar and Priest’s carbon fiber strengths.