Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(||X-[theta]||) where [theta] is an unknown location vector. For p >= 3, conditions on f are given for which there exist minimax estimators [theta](X) satisfying ||Xt|| · ||[theta](X) - X|| <= C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of ||[theta]- [theta]||2. If [theta] is assumed likely to lie in a ball in 1p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.