The Generalized Log-Adjusted Polynomial Family for Reliability and Medical Risk Analysis under Different Non-Bayesian Methods: Properties, Characterizations and Applications

This paper introduces a novel class of continuous probability distributions called the genralized Log-Adjusted Polynomial (GLAP) family, with a focus on the GLAP Weibull distribution as a key special case. The proposed family is designed to enhance the flexibility of classical distributions by incorporating additional parameters that control shape, skewness, and tail behavior. The GLAP Weibull model is particularly useful for modeling lifetime data and extreme events characterized by heavy tails and asymmetry. The paper presents the mathematical formulation of the new family, including its cumulative distribution function, probability density function, and hazard rate function. It also explores structural properties such as series expansions and tail behavior. Risk analysis is conducted using advanced key risk indicators (KRIs), including Value-at-Risk (VaR), Tail VaR (TVaR), and tail mean-variance (TMVq), under various estimation techniques. Estimation methods considered include maximum likelihood (MLE), Cramer–von Mises (CVM), Anderson–Darling (ADE), and their right-tail and left-tail variants. These methods are compared using both simulated and real insurance data to assess their sensitivity to tail events. Finally, the paper provides a comprehensive analysis of risks in the field of reliability and in the medical field. The analysis included examining engineering and medical risks using the aforementioned estimation methods and considering a variety of confidence levels based on five risk measurement and analysis indicators.