Optimal Simultaneous Confidence Bounds

Abstract

The notion of a "simultaneous confidence bound" is redefined by requiring a bound on the expected converge measure (ECM) instead of the coverage probability. This is analogous to a criterion introduced by Spjotvoll for defining simultaneous tests of hypotheses. Bounds which minimize certain width functionals, subject to a bound on the ECM, are characterized. For bounds on a multilinear regression function over an arbitrary subset of Euclidean space, the bounds which minimize weighted average width, among all bounds with prescribed ECM, are expressed in closed form. As a special case, we give a weight function relative to which Scheffe-type bounds are optimal.