A New Light-tailed Weighted Model for Reliability Risk Analysis: Theory, Properties and Case Study

Abstract

This paper introduces the new weighted distribution, an exponential‑type, flexible, right‑skewed, and provably light‑tailed probability model for reliability risk analysis. It is motivated by the fact that failure‑time data are strongly right‑skewed in the body yet exhibit exponential‑type upper tails, so heavy‑tailed models inflate extreme quantiles while simple laws lack body flexibility. It is further motivated by the absence of any weighted extension, unified mathematical theory, or rigorous tail analysis for the Bilal baseline, i.e. the median of three exponential lifetimes (a two‑out‑of‑three system). A third motivation is the literature’s routine advertisement of “heavy‑tailed” constructions without extreme‑value validation, which this paper counters with an honest, mathematically verified tail classification. A consolidated tail theory establishes the Gumbel extreme‑value behavior of the model and explains the drifting, non‑stabilizing patterns of the Hill estimator. A comprehensive risk analysis using Value‑at‑Risk (VaR), Tail Value‑at‑Risk (T‑VaR), Tail Variance (TVc), Tail Mean Variance (TMVc), Peaks Over a Random VaR (PORT‑VaR), and Mean of Order P ( ), validated by tail‑index and extreme value theory (EVT) diagnostics. The results yield a confidence‑level planning ladder for routine maintenance, warranty limits, and redundancy against catastrophic failures, offering engineers a verified light‑tailed alternative to routinely advertised heavy‑tailed constructions.