  We describe a new recentered confidence sphere for the mean, theta, of a
multivariate normal distribution. This sphere is centred on the positive-part
James-Stein estimator, with radius that is a piecewise cubic Hermite
interpolating polynomial function of the norm of the data vector. This radius
function is determined by numerically minimizing the scaled expected volume, at
theta = 0, of this confidence sphere, subject to the coverage constraint. We
use the computationally-convenient formula, derived by Casella and Hwang [3],
for the coverage probability of a recentered confidence sphere. Casella and
Hwang, op. cit., describe a recentered confidence sphere that is also centred
on the positive-part James-Stein estimator, but with radius function determined
by empirical Bayes considerations. Our new recentered confidence sphere
compares favourably with this confidence sphere, in terms of both the minimum
coverage probability and the scaled expected volume at theta = 0.
