Buehler confidence limits and nesting

The Buehler 1 –α upper confidence limit is as small as possible, subject to the constraints that its coverage probability is at least 1 –α and that it is a non‐decreasing function of a pre‐specified statistic T. This confidence limit has important biostatistical and reliability applications. Previous research has examined the way the choice of T affects the efficiency of the Buehler 1 –α upper confidence limit for a given value of α. This paper considers how T should be chosen when the Buehler limit is to be computed for a range of values of α. If T is allowed to depend on α then the Buehler limit is not necessarily a non‐increasing function of α, i.e. the limit is ‘non‐nesting’. Furthermore, non‐nesting occurs in standard and practical examples. Therefore, if the limit is to be computed for a range [α_L, α_U]of values of α, this paper suggests that T should be a carefully chosen approximate 1 –α_L upper limit for θ. The choice leads to Buehler limits that have high statistical efficiency and are nesting.