Approximate Lower Confidence Limits for the Weibull Reliability Function

An iterative technique for obtaining large sample approximate lower s-confidence limits (LCL) for the reliability function of a Weibull probability model is proposed. The approximations are based on the asymptotic distribution of the maximum likelihood (ML) estimator of the reliability function and are valid whenever the ML estimates are obtained from singly censored samples of type I or type II. The iterated approximation is compared with an approximation based on the F distribution and with exact LCLs obtained by Monte Carlo simulation. The main results are: 1. In most circumstances the iterated LCLs are reasonably good approximations of the exact LCLs. 2. At the 95%, and especially at the 90o%, s-confidence levels the iterated approximations are sometimes too high, but never seriously high as is often the case with the non-iterated ML approximation. 3. At the 99% s-confidence level both the iterated and F approximations tend to be low. 4. The F approximation sometimes fails because the degrees of freedom in the approximating F distribution are too low. This happens when the samples are relatively small and the censoring is heavy. 5. The F approximation seems to work best when the estimated reliability is high (> 0.95). When the ML estimator is less than 0.90, the iterated approximation is generally superior. 6. Both approximations improve as the sample size increases and deteriorate as the amount of censoring increases. 7. The iterated approximation does not depend on special tables and is easier to apply than the F approximation.