Better Buehler confidence limits

Consider the reliability problem of finding a 1-[alpha] upper (lower) confidence limit for [theta] the probability of system failure (non-failure), based on binomial data on the probability of failure of each component of the system. The Buehler 1-[alpha] confidence limit is usually based on an estimator of [theta]. This confidence limit has the desired coverage properties. We prove that in large samples the Buehler 1-[alpha] upper confidence limit based on an approximate 1-[alpha] upper limit for [theta] is less conservative, whilst also possessing the desired coverage properties.