A New Model for Reliability Value-at-Risk Assessments with Applications, Different Methods for Estimation, Non-parametric Hill Estimator and PORT-VaRq Analysis

This paper introduces a new extension of the exponential distribution tailored for enhanced reliability and risk analysis. We incorporate several insurance risk indicators like the value-at-risk, tail mean-variance, tail value-at-risk, tail variance, and maximum excess loss to significantly refine reliability risk assessments. These indicators offer vital insights into the financial consequences of extreme risk events and potential for substantial losses. To assess these risk indicators, we explore various non-Bayesian estimation techniques, including maximum likelihood estimation, ordinary least squares estimation, Anderson-Darling estimation, right tail Anderson-Darling estimation, and left tail Anderson-Darling estimation of the second order. Our approach involves a comprehensive simulation study with varying sample sizes, followed by empirical risk analysis using these methods. We also evaluate the applicability of the new model on two real reliability data sets. Finally, we apply the risk indicators including the value-at-risk (VaRq), tail mean-variance (TMVq), tail value-at-risk (TVaRq), tail variance (TVq) and maximum excess loss (MELq) to analyze reliability risk using failure (relief) and survival data. Finally the peaks over a random threshold value-at-risk (PORT-VaRq) analysis under the failure and survival data is presented.