Certified computation of Buehler confidence limits for system reliability

Abstract

Component tests can support a lower confidence limit for a system that has not been tested as a whole, but numerical optimization can compromise its coverage. A feasible point in the minimization defining a Buehler limit gives an upper bound on that limit; reporting it can therefore overstate reliability. We develop a computation that maintains lower and upper bounds on the ideal endpoint for a prescribed ordering. Bounds over entire parameter regions justify the reported lower endpoint at every completed checkpoint, including under adaptive stopping and selection among runs. Separate refinement conditions establish convergence, and an explicit error allocation accounts for simulated tail probabilities and confidence-region localization. The theory retains nuisance and dependence parameters in the observation model. Monotone corner bounds give the implemented specialization for scalar and two-component binomial systems; Bernstein and probability-distance bounds provide further theoretical constructions. Four benchmark problems reach a certified width of $10^{-5}$ under each of four search strategies. Additional stochastic candidates increase cost without reducing subdivisions, whereas complete sample-space enumeration verifies the implemented coverage at three parameter vectors. For the all-success example, the certified lower limit is $0.68128$, compared with $0.41738$ from Bonferroni-adjusted component limits.