A unified minimax result for restricted parameter spaces

We provide a development that unifies, simplifies and extends considerably a number of minimax results in the restricted parameter space literature. Various applications follow, such as that of estimating location or scale parameters under a lower (or upper) bound restriction, location parameter vectors restricted to a polyhedral cone, scale parameters subject to restricted ratios or products, linear combinations of restricted location parameters, location parameters bounded to an interval with unknown scale, quantiles for location-scale families with parametric restrictions and restricted covariance matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ336 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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

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.Comment: Published at http://dx.doi.org/10.1214/074921706000000635 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org