## The Annals of Statistics

### Sufficiency and Invariance in Confidence Set Estimation

Peter M. Hooper

#### 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.

#### Article information

Source
Ann. Statist., Volume 10, Number 2 (1982), 549-555.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345795

Digital Object Identifier
doi:10.1214/aos/1176345795

Mathematical Reviews number (MathSciNet)
MR653529

Zentralblatt MATH identifier
0511.62038

JSTOR