A New Family of Distributions Based on Gamma Frailty

Abstract

This paper proposes a flexible three-parameter lifetime distribution, named the Weibull-Gamma Frailty (W-GF) model, suitable for analyzing survival data influenced by latent or unmeasured factors. The model arises from compounding a Weibull baseline distribution with a gamma-distributed frailty term, allowing it to represent various hazard rate shapes, including increasing, decreasing, and bathtub forms. Analytical expressions for several statistical properties are derived, and multiple estimation approaches are examined, including maximum likelihood, product of spacings, percentile-based, least squares, weighted least squares, and Cramér–von Mises methods. A comprehensive simulation study is conducted to assess the performance of these methods under varying sample sizes. Furthermore, the model is applied to six distinct real-world datasets, consistently demonstrating superior fitting capability compared to existing lifetime models, thereby validating its robustness in capturing heterogeneity in survival data.