## The Annals of Statistics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62A05

Secondary: 62B99: None of the above, but in this section 62F25: Tolerance and confidence regions 62C07: Complete class results

**Keywords**

General MANOVA pivotal quantity randomized set estimator

#### Citation

Hooper, Peter M. Sufficiency and Invariance in Confidence Set Estimation. Ann. Statist. 10 (1982), no. 2, 549--555. doi:10.1214/aos/1176345795. https://projecteuclid.org/euclid.aos/1176345795