A Cox-Weibull Convoluted Proportional Hazard Model with Constrained Parameters for Modelling Short-Term and Long-Term Survival Risks

Survival analysis comprises a set of statistical techniques used to investigate the time duration until occurrence of a specific event.it undergone significant methodological advancement since the introduction of the classical Cox proportional hazards model. The Cox model, by virtue of its semi-parametric formulation, provided researchers with a flexible tool to model time-to-event data without making strict assumptions about the baseline hazard function. However, its flexibility has proven insufficient in many real-world applications, especially where complex hazard dynamics such as dual risk, long-term survivorship, or structural heterogeneity are present, have advanced survival modeling but fall short of addressing the dual risk of hazard: the distinct short-term and long-term risk patterns that may coexist within survival processes. This limitation is particularly problematic in criminological studies of recidivism, where individuals face heightened risk immediately after release, followed by a different pattern of risk in the longer term. Conventional models such as the Cox proportional hazards (PH) model or parametric survival distributions are often too restrictive to capture these complexities. The present study addresses this methodological gap by proposing a convoluted proportional hazard model with constrained parameters. The literature, therefore, has evolved toward mixture cure models, split‐population models, and hybrid specifications that allow simultaneous modeling of short‐term and long‐term risks. Survival analysis has undergone significant methodological advancement but the dual risk of hazard where short-term and long-term risks follow different trajectories remains unaccounted for many approach. This study convoluted proportional hazard model with constrained parameters, This approach integrates the semi-parametric flexibility of the Cox model with the parametric structure of the Weibull distribution, to predicting the probability of recidivism and identify factors that influence the risk of recidivism and study also offers significant improvement by combining Weibull distribution and Cox model by introducing two additional constrained parameters to capture both short term and long term risk of recidivism. The addition of parameter constraints enhances parsimony, stabilty and interpretability. Maximum Likelihood estimation approach was used to estimate the parameter of the newly developed Cox-Weibull model.