Beyond the Tail: Taming Bias in Burr XII Residual Entropy Estimation

Accurately predicting system lifetime is routine in reliability and survival analysis, but quantifying uncertainty in residual life is rarely addressed despite being often more critical. Residual entropy fills this gap by measuring unpredictability in remaining lifetime. Under the flexible Burr XII model, we show that the standard MLE of residual entropy is significantly biased in small samples, limiting its reliability. Through extensive simulation, we compare the MLE with three bias-corrected estimators: Cox–Snell, Firth, and parametric bootstrap. The Cox–Snell correction consistently outperforms, achieving near bias–free estimates and the lowest MSE. We validate this superiority using the Wisconsin Breast Cancer Database, delivering a robust tool for engineers and clinicians to assess survival uncertainty with greater confidence.