GENERALIZATIONS OF JAMES-STEIN ESTIMATORS UNDER SPHERICAL-SYMMETRY

This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions. Specifically, when X approximately f(parallel-to X - theta parallel-to2), we investigate conditions under which estimators of the form X + ag(X) dominate X for loss functions parallel-to-delta - theta-parallel-to 2 and loss functions which are concave in parallel-to-delta - parallel-to-delta2. Additionally, if the scale is unknown we investigate estimators of the location parameter of the form X + aVg(X) in two different settings. In the first, an estimator V of the scale is independent of X. In the second, V is the sum of squared residuals in the usual canonical setting of a generalized linear model when sampling from a spherically symmetric distribution. These results are also generalized to concave loss. The conditions for domination of X + ag(X) are typically (a) parallel-to-g-parallel-to2 + 2-DELTA-degrees g less-than-or-equal-to 0, (b) DELTA-degrees g is superharmonic and (c) 0 < a < 1/pE0(1/parallel-to X parallel-to2), plus technical conditions.