In this article, we demonstrate that Stein estimators of the form [1-a/(b+∥X∥ 2 )]X arise under a broad variety of location parameter problems from a natural calculation, wherein one does not need independence of the coordinates, nor symmetry, and the loss function is quite general, including the important absolute error loss. The results are different in character from classic previous works of Brown, Casella, Strawderman, and Shinozaki. Our method of calculation automatically shows how the Stein estimate could be adapted to a specific parent distribution and a specific loss. A number of examples that illustrate this adaptive calculation result in quite remarkable features. In particular, some very specific and new Stein estimates emerge in important cases, specifically for absolute error loss. The estimates are studied with respect to their risk, both theoretically and via simulation, and the evidence suggests that minimaxity can be expected in generality in 4 or more dimensions. In addition, the specific proposed estimate appears to outperform both X and the ordinary James-Stein estimate under absolute error loss for the normal case.