Beyond Routine Construction: Actuarial Risk Measures, Multi Method Estimation, and Censored Survival Modeling with the Topp–Leone–Heavy Tailed G Family

Abstract

While numerous extensions of the Topp–Leone-G (TL-G) family exist, none simultaneously provide essential contributions that move beyond routine distributional construction. This study introduces the Topp–Leone–heavy‑tailed-G (TL‑HT‑G) family to fill three specific gaps: first, exact closed‑form (up to an infinite series) expressions for four major actuarial risk measures (Value at Risk, Tail Value at Risk, Tail Variance, and Tail Variance Premium); second, a systematic comparison of five estimation methods with formal ranking across sample sizes; and third, validation on both complete and censored real‑world oncological datasets against multiple non‑nested competing models. We derive key theoretical properties including tail asymptotics, identifiability, and analytical hazard rate shapes, and prove that the tail index is directly tunable via a single parameter—offering explicit control over tail heaviness not available in standard TL‑G extensions. A large‑scale Monte Carlo simulation (3,000 replications; sample sizes 25 to 800) reveals that maximum likelihood estimation consistently achieves the lowest cumulative rank across all metrics. Numerical evaluations of actuarial risk measures confirm that the TL‑HT‑W distribution exhibits substantially heavier tails than Kumaraswamy‑Weibull, type‑I heavy‑tailed Weibull, and standard Weibull benchmarks. In four real‑data scenarios (complete and censored leukemia and bladder cancer data), the TL‑HT‑W model outperforms six non‑nested heavy‑tailed competitors, yielding the lowest goodness‑of‑fit values, the highest Kolmogorov–Smirnov p‑values, and excellent alignment with empirical cumulative distribution and Kaplan–Meier curves. Collectively, these features move the TL‑HT‑G family beyond routine construction, establishing it as a theoretically grounded, empirically superior, and ready‑to‑use model for high‑stakes applications in actuarial science, clinical survival analysis, and reliability engineering.