  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.
