Statistica Sinica 5(1995), 291-301

COMPLETE CLASSES FOR CONFIDENCE SET ESTIMATION

L. D. Brown and Arthur Cohen

University of Pennsylvania and Rutgers University

Abstract: Consider the statistical model where a statistic T  has a distribution belonging to a one parameter family which has the strict monotone likelihood ratio property. We study confidence set estimation for two types of risk functions. One is a vector risk consisting of two components: one is one minus the coverage probability and the other is the expected measure of the set. The other risk is a linear combination of the two components. Under certain conditions we find that monotone procedures are a complete class and also that procedures with an interval property are a complete class. As consequences we find that nonrandomized procedures are a complete class. Furthermore, we find an analogue to the Rao-Blackwell theorem and construction for exponential family models. Other applications and observations are noted.

Key words and phrases: Admissibility, interval property, monotone procedure, randomized procedures, Rao-Blackwell construction.