Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p >= 4) spherically symmetric (s.s.) distribution about [theta], minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L([delta], [theta]) = ([delta] - [theta])' D([delta] - [theta]) where D is a known p - p positive definite matrix. For C a p - p known positive definite matrix, conditions are given under which estimators of the form [delta]a,r,C,D(X) = (I - (ar(X2)) D-1/2CD1/2 X-2)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X1, X2, ..., Xn are taken on a p-dimensional s.s. distribution about [theta], any spherically symmetric translation invariant estimator, [delta](X1, X2, ..., Xn), with have a s.s. distribution about [theta]. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.