Distributional Analysis and Risk Assessment of U.K. Motor Non-Comprehensive Claims Using the Log-Exponential Family with Properties and Characterizations

This paper studies the log-exponential-exponential (LEE) distribution which is a novel special case of the logexponential G (LE) family, tailored for flexible modeling of insurance claim sizes. The LEE distribution demonstrates exceptional versatility in capturing diverse density shapes including light-tailed with different forms, whose sign determinesthe direction of skewness. We derive explicit expressions for its probability density function and establish rigorouscharacterizations using truncated moments and reverse-hazard rate identities. A comprehensive simulation study is conductedto assess the performance of six estimation techniques: maximum likelihood estimation (MLE), ordinary least squares (OLS),Cramer–von Mises estimation (CVME), Anderson–Darling estimation (ADE), right-tail Anderson–Darling estimation ´(RTADE), and left-tail Anderson–Darling estimation (LTADE), across various parameter configurations and sample sizes.Finally, we compute key risk indicators (KRIs) including Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Tail Variance(TV), Tail Mean–Variance (TMV), and Expected Loss (EL) using all six estimation methods, applied to real U.K. motornon-comprehensive claims triangle data