On the estimation of a normal variance

Complementing Rukhin's [5] investigations, the mean squared error of members of a family of preliminary-test estimators and members of the basic family of shrinkage estimators are compared. The reference point is Stein's [10] estimator that is contained in both classes. It turns out that Stein's estimator has certain qualitative optimality properties among the preliminary-test estimators. But it also turns out that, while Rukhin's 4% maxim in the minimax shrinkage class is well known and frequently cited, without minimaxity greater improvements can be achieved in his local sense by estimators coming from the preliminary-test class. This underlines another aspect of Stein's original insight. Most of our findings are presented graphically.