On the optimality and limitations of Buehler bounds

Abstract

In 1957, R.J. Buehler gave a method of constructing honest upper confidence limits for a parameter that are as small as possible subject to a pre-specified ordering restriction. In reliability theory, these 'Buehler bounds' play a central role in setting upper confidence limits for failure probabilities. Despite their stated strong optimality property, Buehler bounds remain virtually unknown to the wider statistical audience. This paper has two purposes. First, it points out that Buehler's construction is not well defined in general. However, a slightly modified version of the Buehler construction is minimal in a slightly weaker, but still compelling, sense. A proof is presented of the optim..