Optimal Confidence Interval for a Ranked Parameter

There are given k univariate distributions, indexed by a real-valued parameter θ, and k independent observations, one from each distribution. Let θ* denote the largest among the values of θ associated with the given distributions. This article is concerned with the estimation of θ*. A class of confidence intervals and a subclass of “optimal” confidence intervals for θ* are given. For a general class of distributions it is shown that the coverage probability of the confidence intervals is minimized for a specified configuration of the set of admissible values of θ.