## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Number 50, 2006, 112-126

### On the behavior of Bayesian credible intervals for some restricted parameter space problems

Éric Marchand and William E. Strawderman

#### 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.

#### Chapter information

**Source***Recent Developments in Nonparametric Inference and Probability: Festschrift for Michael Woodroofe* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196284056

**Digital Object Identifier**

doi:10.1214/074921706000000635

**Mathematical Reviews number (MathSciNet)**

MR2409067

**Zentralblatt MATH identifier**

1268.62033

**Subjects**

Primary: 62F10: Point estimation 62F15: Bayesian inference 62F30: Inference under constraints 62F99: None of the above, but in this section

**Keywords**

restricted parameter space Bayesian methods confidence intervals frequentist coverage probability unimodality

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Marchand, Éric; Strawderman, William E. On the behavior of Bayesian credible intervals for some restricted parameter space problems. Recent Developments in Nonparametric Inference and Probability, 112--126, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000635. https://projecteuclid.org/euclid.lnms/1196284056