A New Version of the Compound Quasi-Lomax Model: Properties, Characterizations and Risk Analysis under the U.K. Motor Insurance Claims Data

Abstract

This paper introduces a new lifetime distribution, the Compound Quasi-Lomax (CQLx) model, designed to enhance the modeling of heavy-tailed data in actuarial and financial risk analysis. The CQLx distribution is developed through a novel extension of the Lomax family, offering increased flexibility in capturing extreme values and complex data behaviors. Key mathematical properties are derived. Characterization of the model is achieved via truncated moments and the reverse hazard function. Several estimation methods are employed including the Maximum Likelihood Estimation (MLE), Cramér-von Mises (СУМ), Anderson-Darling Estimation (ADE), Right-Tail Anderson-Darling Estimation (RTADE), and Left-Tail Anderson-Darling Estimation (LTADE). A comprehensive simulation study evaluates the performance of these methods in terms of bias and root mean square error (RMSE) across various sample sizes. Risk measures such as Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Tail Variance (TV), Tail Mean Variance (TMV), and Expected Loss (EL) are computed under artificial and real financial insurance claims data. The results demonstrate that MLE generally provides the most accurate and stable estimates, particularly for larger samples, while CVM and ADE tend to overestimate risk, especially at higher quantiles. The CQLx model shows superior performance in fitting extreme claim-size data, making it a robust tool for risk management.