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VOL. 50 | 2006 On the behavior of Bayesian credible intervals for some restricted parameter space problems
Éric Marchand, William E. Strawderman

Editor(s) Jiayang Sun, Anirban DasGupta, Vince Melfi, Connie Page

Abstract

For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1 -\alpha)\%$ HPD credible sets associated with priors obtained as the truncation of noninformative priors onto the restricted parameter space. Namely, they establish the attractive lower bound of $\frac{1-\alpha}{1+\alpha}$ for the frequentist coverage probability of these procedures. In this work, we establish that the lower bound of $\frac{1-\alpha}{1+\alpha}$ is applicable for a substantially more general setting with underlying distributional symmetry, and obtain various other properties. The derivations are unified and are driven by the choice of a right Haar invariant prior. Investigations of non-symmetric models are carried out and similar results are obtained. Namely, (i) we show that the lower bound $\frac{1-\alpha}{1+\alpha}$ still applies for certain types of asymmetry (or skewness), and (ii) we extend results obtained by Zhang and Woodroofe (2002) for estimating the scale parameter of a Fisher distribution; which arises in estimating the ratio of variance components in a one-way balanced random effects ANOVA. Finally, various examples illustrating the wide scope of applications are expanded upon. Examples include estimating parameters in location models and location-scale models, estimating scale parameters in scale models, estimating linear combinations of location parameters such as differences, estimating ratios of scale parameters, and problems with non-independent observations.

Information

Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1268.62033
MathSciNet: MR2409067

Digital Object Identifier: 10.1214/074921706000000635

Subjects:
Primary: 62F10 , 62F15 , 62F30 , 62F99

Keywords: Bayesian methods , confidence intervals , frequentist coverage probability , restricted parameter space , Unimodality

Rights: Copyright © 2006, Institute of Mathematical Statistics

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