On the Optimality and Limitations of Buehler Bounds

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 optimality of this modified Buehler construction under minimal regularity conditions. Second, the paper demonstrates that Buehler bounds can be expressed as the supremum of Buehler bounds conditional on any nuisance parameters, under very weak assumptions. This result is then used to demonstrate that Buehler bounds reduce to a trivial construction for the location-scale model. This places important practical limits on the application of Buehler bounds and explains why they are not as well known as they deserve to be. Copyright 2003 Australian Statistical Publishing Association Inc..