Gaussian-Bootstrap-Tuned Logit-Shrinkage Estimation of the Log-Logistic Reliability Function under Right Censoring and Precautionary Loss

Abstract

Reliability estimates from small right-censored lifetime samples can be unstable when both parameters of the log-logistic model are estimated. This study shrinks the logit of the mission-time reliability toward an external target and selects the shrinkage coefficient by minimizing a deterministic stratified Gaussian approximation to precautionary risk. The method is compared primarily with the classical plug-in estimator and a positive-part adaptive logit rule, while fixed shrinkage rules are retained as secondary oracle/reference comparators. The Monte Carlo study used 10,000 replications per core condition. With an exact target, the proposed estimator achieved relative risks of approximately 2.09–2.15 against the MLE. Its benefit was conditional on target quality: in the baseline setting, a local sensitivity study placed the approximate break-even target-shift interval at−0.081 <R0− R<0.122. Under exponential random censoring, relative risk remained about 2.10. A quadrature-size study showed negligible change between B= 500 and B= 1000, and a computational benchmark found the stratified Gaussian tuning close to random-Gaussian and nonparametric pair-bootstrap tuning but substantially faster because it avoids repeated model refitting. In 500 repeated heart-transplant pilot/main splits, the internally estimated  but it should use k= 1 and reduce to the MLE when no defensible target is available; it is not uniformly superior under target misspecification.