Estimation of the mean of a spherically symmetric distribution with constraints on the norm

The authors consider the problem of estimating, under quadratic loss, the mean of a spherically symmetric distribution when its norm is supposed to be known and when a residual vector is available. They give a necessary and sufficient condition for the optimal James-Stein estimator to dominate the usual estimator. Various examples are given that are not necessarily variance mixtures of normal distributions. Consideration is also given to an alternative class of robust James-Stein type estimators that take into account the residual vector. A more general domination condition is given for this class.