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June, 1982 Sufficiency and Invariance in Confidence Set Estimation
Peter M. Hooper
Ann. Statist. 10(2): 549-555 (June, 1982). DOI: 10.1214/aos/1176345795


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.


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Peter M. Hooper. "Sufficiency and Invariance in Confidence Set Estimation." Ann. Statist. 10 (2) 549 - 555, June, 1982.


Published: June, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0511.62038
MathSciNet: MR653529
Digital Object Identifier: 10.1214/aos/1176345795

Primary: 62A05
Secondary: 62B99 , 62C07 , 62F25

Keywords: General MANOVA , pivotal quantity , randomized set estimator

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • June, 1982
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