An Innovated G Family: Properties, Characterizations and Risk Analysis under Different Estimation Methods

This paper introduces a novel class of continuous probability distributions called the Log-Adjusted Polynomial (LAP) G family, with a focus on the LAP 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 LAP Weibull model is particularly useful for modeling lifetime data, extreme events, and insurance claims 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 risk measures, 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-world insurance data to assess their sensitivity to tail events. The performance of each estimator is evaluated in terms of bias, accuracy, and robustness in capturing extreme risks. The LAP Weibull model demonstrates superior performance in fitting heavy-tailed data compared to traditional models. AD2LE emerges as the most risk-sensitive estimator, producing the highest values for all key risk indicators. ADE also performs well, offering a balance between sensitivity and stability. MLE and CVM tend to underestimate tail risks, which could lead to insufficient capital reserves in insurance applications. The study highlights the importance of selecting appropriate estimation techniques based on the specific goals of the risk analysis. With its enhanced flexibility and performance in modeling extreme risks, the LAP Weibull model offers a robust framework for modern risk assessment. The findings support the use of AD2LE or ADE in high-stakes risk management scenarios, especially when dealing with heavy-tailed insurance data. This work contributes to the growing literature on advanced statistical models for actuarial and financial risk analysis. The LAP Weibull model proves particularly useful in capturing the tail behavior of claim distributions, improving the accuracy of risk predictions. The paper provides a solid foundation for future applications of the LAP family in modeling complex real-world phenomena under uncertainty.