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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