A Novel Hybrid of Weibull-Exponential-Gamma (W-E-G) Distribution with Applications to Bladder Cancer Data

Abstract

The Weibull-Exponential-Gamma (WEG) hybrid distribution is introduced for analyzing cancer survival data. The proposed distribution combines the flexibility of the Weibull and Gamma distributions with the simplicity of the Exponential model, providing a unified framework capable of accommodating increasing, decreasing, and non-monotonic hazard functions. The probability density function, cumulative distribution function, and key statistical properties-including moments, hazard function, and reliability characteristics-are derived and analyzed to understand the distribution’s behavior under varying parameter configurations. A Monte Carlo simulation with 1,000 replicates across sample sizes of 50 to 1,000 demonstrated that Bias, MSE, and RMSE approached zero as sample size grew, confirming estimator consistency. When applied to real-world cancer survival data, the WEG model outperformed Exponential (LL = -382.14, AIC = 766.28), Weibull, Gamma, Weibull-Gamma, and Exponential-Gamma distributions, achieving the highest LL (-361.42) and lowest AIC (732.85) and BIC (748.49). A clinical surveillance schedule derived from the WEG hazard function stratified post-treatment risk into six phases: very high (0-3 months, h = 0.14-0.10) requiring monthly visits; high (3–6 months, h = 0.10-0.08); moderate (6-12 months, h = 0.08-0.05); low (12-24 months, h = 0.05-0.03); very low (24-36 months, h = 0.03-0.02); and minimal (>36 months, h < 0.02) needing annual follow-up. The WEG distribution offers a flexible, evidence-based tool for bladder cancer survival modeling and risk-adapted patient monitoring. Based on these findings, the WEG distribution is recommended as a flexible and robust tool for survival analysis, particularly in biomedical research where heterogeneity and non-standard hazard behaviors are common.