Sufficiency and Invariance in Confidence Set Estimation

Abstract

This paper describes how sufficiency and invariance considerations can be applied in problems of confidence set estimation to reduce the class of set estimators under investigation. Let $X$ be a random variable taking values in $\mathscr{X}$ with distribution $P_\theta, \theta \in \Theta$, and suppose a confidence set is desired for $\gamma = \gamma(\theta)$, where $\gamma$ takes values in $\Gamma$. The main tools used are (i) the representation of randomized set estimators as functions $\varphi: \mathscr{X} \times \Gamma \rightarrow \lbrack 0,1 \rbrack$, and (ii) the definition of sufficiency in terms of a certain family of distributions on $\mathscr{X} \times \Gamma$. Sufficiency and invariance reductions applied in tandem to $\mathscr{X} \times \Gamma$ yield a class of set estimators that is essentially complete among all invariant set estimators, provided the risk function depends only on $E_{\theta \varphi} (X, \gamma), (\theta, \gamma) \in \Theta \times \Gamma$. Several illustrations are given.