  Casella and Hwang, 1983, JASA, introduced a broad class of recentered
confidence spheres for the mean $\boldsymbol{\theta}$ of a multivariate normal
distribution with covariance matrix $\sigma^2 \boldsymbol{I}$, for $\sigma^2$
known. Both the center and radius functions of these confidence spheres are
flexible functions of the data. For the particular case of confidence spheres
centered on the positive-part James-Stein estimator and with radius determined
by empirical Bayes considerations, they show numerically that these confidence
spheres have the desired minimum coverage probability $1-\alpha$ and dominate
the usual confidence sphere in terms of scaled volume. We shift the focus from
the scaled volume to the scaled expected volume of the recentered confidence
sphere. Since both the coverage probability and the scaled expected volume are
functions of the Euclidean norm of $\boldsymbol{\theta}$, it is feasible to
optimize the performance of the recentered confidence sphere by numerically
computing both the center and radius functions so as to optimize some clearly
specified criterion. We suppose that we have uncertain prior information that
$\boldsymbol{\theta}= \boldsymbol{0}$. This motivates us to determine the
center and radius functions of the confidence sphere by numerical minimization
of the scaled expected volume of the confidence sphere at $\boldsymbol{\theta}=
\boldsymbol{0}$, subject to the constraints that (a) the coverage probability
never falls below $1-\alpha$ and (b) the radius never exceeds the radius of the
standard $1-\alpha$ confidence sphere. Our results show that, by focusing on
this clearly specified criterion, significant gains in performance (in terms of
this criterion) can be achieved. We also present analogous results for the much
more difficult case that $\sigma^2$ is unknown.
