Invariant confidence sets with smallest expected measure
type: Item
instance of: scholarly article
title: Invariant confidence sets with smallest expected measure
zbMATH Open document ID: 0536.62024
author: Peter M. Hooper
published in: The Annals of Statistics
publication date: 1982-01-01T00:00:00Z
review text: A confidence set C(X), \(X\in {\mathcal X}\), is wanted for a function \(\gamma =\gamma(\theta)\in \Gamma\) of a parameter \(\theta\in \Theta\) that subject to covering the true value of \(\gamma\) with probability at least 1- \(\alpha\) (for every \(\theta\in \Theta)\) in some sense minimizes the probabilities of covering wrong values of \(\gamma\). This is made precise by adopting a measure \(m(\cdot,\theta)\) on \(\Gamma\), depending on \(\theta\), and defining the expected measure as \(\int P_{\theta}(\gamma \in C(X)) m(d\gamma,\theta)\). (Actually, this expression is modified slightly to allow for randomization.) It is assumed that there is an invariance group that acts on \({\mathcal X}\) and on \(\Gamma\), and acts transitively on \(\Theta\). If m is chosen to be equivariant and if the confidence set is restricted to be invariant, then the confidence set minimizing the expected measure (subject to the coverage requirement) is independent of \(\theta\). If an invariantly sufficient statistic on \({\mathcal X}\times \Gamma\) exists, then the solution can be expressed explicitly in terms of a Neyman-Pearson-type rule. For measures m that are invariant the minimizing invariant confidence set is minimax among all confidence sets provided conditions similar to those in the Hunt-Stein theorem are satisfied. Several examples are presented, drawn from multivariate analysis, location and scale parameter families, and nonparametrics. It is pointed out that for the purpose of generating simultaneous confidence sets of families of parametric functions of \(\gamma\) several of the confidence sets for \(\gamma\) in the examples have the drawback of not being exact in the reviewer's sense [Multivariate Analysis V, Proc. 5th int. Symp., Pittsburgh 1978, 483-498 (1980; Zbl 0431.62031)]. A few minor corrections to the paper under review have appeared in Ann. Stat. 12, 784 (1984).
Mathematics Subject Classification ID: 62F25
Mathematics Subject Classification ID: 62H12
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 62C20
zbMATH DE Number: 3852210
zbMATH Keywords: invariant confidence sets
zbMATH Keywords: smallest expected measure
zbMATH Keywords: fiducial, GMANOVA
zbMATH Keywords: pivotal quantity
zbMATH Keywords: MANOVA
zbMATH Keywords: equivariance property
zbMATH Keywords: randomization
zbMATH Keywords: invariantly sufficient statistic
zbMATH Keywords: Neyman-Pearson-type rule
zbMATH Keywords: Hunt-Stein theorem
zbMATH Keywords: examples
zbMATH Keywords: simultaneous confidence sets
reviewed by: Robert A. Wijsman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176345994
OpenAlex ID: W2051758875
DOI: 10.1214/AOS/1176345994
description: scientific article; zbMATH DE number 3852210