Improved estimates of location in the presence of an unknown scale

We investigate conditions under which estimators of the form X + aU'Ug(X) dominate X when X, a p - 1 vector, and U, an m - 1 vector, are distributed such that [X1, X2,..., Xp, U1, U2,..., Up]'/[sigma] has a spherically symmetric distribution about [[theta]1, [theta]2,..., [theta]p, 0, 0,..., 0]', where [sigma] is an unknown scale. Brandwein and Strawderman [2] have results for quadratic loss and we extend these results to concave functions of quadratic loss and to general quadratic loss.