Shrinkage estimation towards a closed convex set with a smooth boundary

We give James-Stein type estimators of a multivariate normal mean vector by shrinkage towards a closed convex set K with a smooth or piecewise smooth boundary. The rate of shrinkage is determined by the curvature of the bound ary of K at the projection point onto K. By considering a sequence of polyt opes K-j converging to K, we show that a particular estimator we propose is the limit of a sequence of shrinkage estimators towards K-j given by M. E. Beck (1982). In fact our estimators reduce to the James-Stein estimator an d to the Beck estimator when K is a point and a convex polyhedron, respecti vely. Therefore they can be considered as natural extensions of these estim ators. Furthermore we apply the same method to the problem of improving the restricted mle by shrinkage towards the origin in the multivariate normal mean model where the mean vector is restricted to a closed convex cone with a smooth or piecewise smooth boundary. We demonstrate our estimators in tw o settings, one shrinking to a ball and the other shrinking to the cone of nonnegative definite matrices. (C) 2000 Academic Press.