The Double-log-exponential Family for Risk Analysis: Properties, Characterizations and Application under the U.K. Motor Non-Comprehensive Claims Triangle Data

This paper introduces the Double-Log-Exponential-Exponential (DLEE) distribution, a new special case of thedouble-log-exponential G (DLEG) family, designed for flexible modeling of insurance claim sizes. The DLEE exhibits remarkable adaptability in density shapes including left-skewed, bimodal, and light-tailed configurations via a single shape parameter θ, whose sign governs skewness direction. We derive some expressions for its density, and provide rigorous characterizations based on truncated moments and reverse-hazard identities. A comprehensive simulation study evaluates six estimation methods namely, maximum likelihood estimation (MLE), ordinary least squares (OLS), Cramer–von Misesestimation (CVME), Anderson–Darling estimation (ADE), right-tail Anderson–Darling estimation (RTADE), and left-tail Anderson–Darling estimation (LTADE)) across multiple parameter scenarios and sample sizes. Finally, the estimated Key Risk Indicators (KRIs), namely Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Tail Variance (TV), Tail Mean–Variance (TMV), and Expected Loss (EL), under the six estimation methods applied to the real U.K. motor non-comprehensive claims triangle.