Interval Estimation of Stress–Strength Reliability R=P(X<Y) for Birnbaum–Saunders Distribution

Stress–strength reliability, $R=P(X<Y)$, is the probability that the strength Y of a component exceeds the stress X, and is widely used in engineering and medicine. The Birnbaum–Saunders (BS) distribution is well suited to positive, right-skewed data. Existing work on R for two independent BS variables has either assumed equal shape parameters or not evaluated interval estimators. Four intervals for R are constructed without parameter restrictions: the generalized confidence interval (GCI), the generalized fiducial confidence interval (GFCI), the Bayesian credible interval (BCrI) and the highest posterior density (HPD) interval. These are compared with three likelihood-based intervals: the asymptotic confidence interval (ACI), its logit-transformed version (ACI-logit) and the bootstrap percentile confidence interval (BPCI). R is computed from a one-dimensional integral that is accurate, invariant to the units of the data and much faster than the exact expression. All seven are assessed by simulation with equal and unequal sample sizes. The results suggest the GCI for small samples, as its coverage probability stayed close to or above the nominal level in all settings. When both samples contain at least 30 observations, the BCrI and the ACI-logit give adequate coverage with generally shorter intervals. Two real-world datasets illustrate the methods.