  For normal canonical models, and more generally a vast array of general
spherically symmetric location-scale models with a residual vector, we consider
estimating the (univariate) location parameter when it is lower bounded. We
provide conditions for estimators to dominate the benchmark minimax MRE
estimator, and thus be minimax under scale invariant loss. These minimax
estimators include the generalized Bayes estimator with respect to the
truncation of the common non-informative prior onto the restricted parameter
space for normal models under general convex symmetric loss, as well as
non-normal models under scale invariant $L^p$ loss with $p>0$. We cover many
other situations when the loss is asymmetric, and where other generalized Bayes
estimators, obtained with different powers of the scale parameter in the prior
measure, are proven to be minimax. We rely on various novel representations,
sharp sign change analyses, as well as capitalize on Kubokawa's integral
expression for risk difference technique. Several other analytical properties
are obtained, including a robustness property of the generalized Bayes
estimators above when the loss is either scale invariant $L^p$ or asymmetrized
versions. Applications include inference in two-sample normal model with order
constraints on the means.
