Competing Risks with Common Shocks: Joint Survival, Copulas, Censoring, Frailty, and Marshall–Olkin Models

This study examines likelihood-based estimation of the joint survival function S(t1,t2)=Pr{T(1)>t1,T(2)>t2} for systems with two competing failure modes observed under right censoring. Rather than introducing a new distributional family, the study compares established dependence mechanisms within a common observed-data framework. Exponential and Weibull margins are combined with three types of dependence: Archimedean copulas, represented by the Gumbel and Clayton families; shared gamma frailty, used to model latent measurement-level heterogeneity; and Marshall–Olkin extensions, used to represent common shocks and simultaneous failures. The same observation scheme, likelihood construction, censoring design, and performance criteria are used across models. Model performance is evaluated through Monte Carlo simulation using bias, integrated mean squared error, and empirical coverage, and the workflow is illustrated with the Device G reliability data. The results show that ignoring dependence can distort joint survival estimates, especially under moderate or high censoring. They also show that copula, frailty, and Marshall–Olkin specifications can lead to different reliability assessments because they encode different stochastic mechanisms. The estimation workflow includes multi-start optimization and diagnostics for boundary solutions, Hessian stability, and irregular likelihood behavior.