Optimal Fixed Size Confidence Procedures for a Restricted Parameter Space

Abstract

Optimal fixed size confidence procedures are derived for the mean of a normal random variable with known variance, when the mean is restricted to a compact interval. These confidence procedures are, in turn, based on the solution of a related minimax decision problem which is characterized by a zero-one loss function and a compact interval parameter space. The minimax rules obtained are nonrandomized, admissible, Bayes procedures. The decision-theoretic results are extended in two ways: (i) structurally similar (admissible) Bayes minimax rules are also obtained when the sampling distribution has a density function which is unimodal, symmetric about the location parameter and possesses a (strictly) monotone likelihood ratio; (ii) structurally similar minimax rules (minimax within the class of nonrandomized, odd, monotone procedures) are again obtained when the assumption of a monotone likelihood ratio is relaxed.