We study the problem of constructing confidence sets for the mean vector θ of a k-variate spherically symmetric distribution by centring them at the positive-part James–Stein estimator. Exploiting its superior risk properties whenever k ≥ 3, we first derive in Chapter 2 the exact sampling law of the test statistic T (X ) = ∥ ˆθ+ J S − θ∥2, showing it consists of a point mass at ∥θ∥2 and a continuous density component valid for any spherically symmetric model. In Chapter 3, we develop level-α procedures for both simple and composite hypotheses about θ, illustrated by a worked example with k = 4, α = 0.05. Chapter 4 then inverts these tests to form (1 − α)-confidence sets via two approaches: (i) a plug-in method using various norm estimators (including the James–Stein shrinkage itself) and (ii) a test-inversion principle guaranteeing exact coverage. Numerical comparisons confirm that the plug-in method produces smaller radii than classical sample-mean–centred sets. Our work thus extends classical multivariate inference by integrating shrinkage estimation into confidence-set theory for spherical distributions.