Optimal Confidence Sets, Bioequivalence, and the Limaçon of Pascal

Abstract

We begin with a decision-theoretic investigation into confidence sets that minimize expected volume at a given parameter value. Such sets are constructed by inverting a family of uniformly most powerful tests, and hence they also enjoy the optimality property of being uniformly most accurate. In addition, these sets possess Bayesian optimal volume properties and represent the first case (to our knowledge) of a frequentist 1 – α confidence set that possesses a Bayesian optimality property. The hypothesis testing problem that generates these sets is similar to that encountered in bioequivalence testing. Our sets are optimal for testing bioequivalence in certain settings; in the case of the normal distribution, the optimal set is a curve known as the limaçon of Pascal. We illustrate the use of these curves with a biopharmaceutical example.