This paper introduces the logarithmic Topp–Leone-G (LTL-G) family, a novel generalized class engineered to significantly enhance the modeling capacity of baseline distributions for complex empirical data exhibiting pronounced skewness, heavy tails, and non-monotonic hazard structures. We rigorously establish the theoretical foundations of the proposed specification, deriving explicit linear expansions for the probability density function, closed-form expressions for ordinary and incomplete moments, and comprehensive distributional characterizations based on truncated moments, reverse hazard functions, and conditional expectations. To critically evaluate inferential reliability, we designed an extensive Monte Carlo simulation protocol comparing six competing estimation methodologies including the maximum likelihood (MLE), ordinary least squares (OLS), Cramér–von Mises (CVM), Anderson–Darling (ADE), right-tail ADE (RTADE), and left-tail ADE (LTADE) across systematically varied sample sizes and challenging parameter configurations. The finite-sample performance is rigorously quantified through bias, root mean squared error, and Kolmogorov–Smirnov diagnostics, with dedicated attention to the convergence behavior of key risk indicators (KRIs). We compute Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Tail Variance (TV), Tail Mean Variance (TMV), and expected shortfall (ELq) to demonstrate the framework’s superior capacity for extreme-tail quantification under data scarcity. Empirical validation is conducted using two high-stake real-life datasets: Social Security Administration (SSA) disability beneficiary records and a UK motor non-comprehensive claims development triangle. The analytical results consistently reveal that the new family specification accurately accommodates extreme dispersion and temporal claim dependencies, while delivering a statistically rigorous foundation for modern actuarial reserving and evidence-based capital allocation.
Ibrahim et al. (Wed,) studied this question.