A New Lindley Frailty Model with Validations, Medical Censored Applications and Value-at-Risk Analysis

Abstract

This paper introduces a novel two-parameter Lindley frailty (2PLF) model designed for enhanced risk analysis and survival modeling. The proposed model incorporates an additional shape parameter, offering increased flexibility in capturing varying tail behaviors and latent heterogeneity in time-to-event data. To estimate the parameters of the 2PLF model, several methods are considered, including maximum likelihood estimation (MLE), Cram´er–von Mises (CVM), Anderson--Darling (ADE), and improved variants such as right-tail ADE (RTADE) and left-tail ADE (LEADE). These approaches are rigorously evaluated through simulation studies under different sample sizes and censoring levels to assess their performance using metrics such as bias, root mean squared error (RMSE), mean absolute deviation (Dabs), and maximum absolute deviation (Dmax). The study further investigates the application of the 2PLF model in real applications by analyzing datasets from emergency care and insurance claims, demonstrating its practical utility in risk assessment and decision-making. Risk indicators such as Value-at-Risk (VaR), Tail VaR (TVaR), and tail mean-variance are employed to evaluate the model's predictive accuracy and robustness. The results confirm that the 2PLF model outperforms traditional models, particularly in scenarios involving complex hazard patterns and high-frailty conditions. This work bridges a critical research gap by providing theoretical insights, empirical validation, and practical implementation of the 2PLF model, making it a valuable tool for researchers and practitioners in fields such as actuarial science, epidemiology, engineering, and survival analysis.