Minimax Estimation of Location Parameters for Spherically Symmetric Unimodal Distributions Under Quadratic Loss

Families of minimax estimators are found for the location parameter of a p-variate (pgt; or = 3) spherically symmetric unimodal(s.s.u.)distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by L(delta,theta) = (delta - theta)'D(delta - theta) where D is a known pxp positive definite matrix, one main result, for one observation, X, on a multivariate s.s.u. distribution about theta, presents a class of minimax estimators whose risks dominate the risk of X, provided pgt; or = 3 and trace D gt; or equal 2dL where dL is the maximum eigenvalue of D. This class is given by Delta a,r(X)=(1-a(r(||X||2)/||X||2))X where 0lt; or = r(.)lt; or = 1, r(||X||2) is nondecreasing, r(||X||2)/||X||2 is nonincreasing, and 0lt;alt;(co/E(||X||-2)((traceD/dL)-2), where co=2p/((p+2)(p-2)) when pgt; or = 4 and co = .96 when p=3