Exact Beta-P Margins for Dependent Competing Risks: Partial Identification and Sensitivity Bounds

Competing-risks data identify the earliest failure time and its cause, but generally do not identify the dependence among latent failure times. We construct a shared-gamma frailty model that preserves distinct Beta-P marginal survival distributions exactly. Within this model, we derive the joint survival function, cause-specific subdensities, and net, crude, and partial-crude risk functionals. We establish positive upper-orthant dependence and an ordering result for nested active-risk sets. We also define model-specific structural sensitivity measures and likelihood-supported probability regions. Using a publicly available transplantation dataset (n = 177), we compare the Aalen–Johansen estimates with the independent Beta-P boundary model and examine dependence through a multi-start profile analysis that re-estimates the marginal parameters at each of 25 grid values. The best likelihood found occurs at the independence boundary (log-likelihood approximately −561.2107); the numerical search found no improvement for positive dependence. We report descriptive profile-grid probability ranges at 12 and 24 months and separate fixed-margin sensitivity ranges. Seeded Monte Carlo studies check the probability calculations and the complete estimator. The formula check has a maximum absolute discrepancy of 0.0016; a separate 300-replicate study obtains reproducible constrained optimization but reveals substantial boundary selection and weak complete-estimator identification at n = 177. The profile-grid ranges are neither confidence intervals nor certified sharp bounds. The model illustrates how assumptions about latent dependence affect latent and observable risk functionals; it does not demonstrate improved clinical prediction.