A Proposed Odd Rayleigh-Exponential Distribution with Shared Gamma Frailty Survival Model

Clustered survival data frequently exhibit unobserved heterogeneity that violates the independence assumption underlying conventional proportional hazards models, resulting in inefficient estimation and potentially misleading inference. This study develops a shared Gamma frailty model based on the Odd Rayleigh–Exponential Distribution (OR-ED) to accommodate latent cluster-level dependence in time-to-event data. The proposed framework extends the OR-ED to a proportional hazards frailty setting by deriving closed-form expressions for the marginal joint survival function and marginal likelihood through the Laplace transform of the Gamma frailty distribution. The corresponding score equations were derived, and model parameters were estimated using the maximum likelihood estimation method. The proposed model is assessed through Monte Carlo simulation experiments under varying cluster sizes and censoring levels and is compared with the corresponding OR-ED model that ignores frailty. The simulation results demonstrate that incorporating shared frailty substantially improves parameter estimation, produces hazard functions that closely follow the underlying data-generating mechanism, and provides more reliable inference than the no-frailty counterpart when unobserved heterogeneity is present. Graphical diagnostics further support the adequacy of the Gamma frailty assumption in representing latent cluster effects. Overall, the proposed OR-ED shared Gamma frailty model provides a tractable and flexible framework for analysing clustered survival data and offers an effective alternative for applications involving correlated failure times and unobserved heterogeneity.