James-Stein Type Estimator by Shrinkage to Closed Convex Set with Smooth Boundary

We give James-Stein type estimators of multivariate normal mean vector by shrinkage to closed convex set K with smooth or piecewise smooth boundary. The rate of shrinkage is determined by the curvature of boundary of K at the projection point onto K. By considering a sequence of polytopes Kj converging to K, we show that a particular estimator we propose is the limit of a sequence of estimators by shrinkage to Kj given by Bock (1982). In fact our estimators reduce to the polyhedron, respectively. Therefore they can be considered as natural extensions of these estimators. Furthermore we apply the same method to the problem of improving the restricted mle by shrinkage toward the origin in the multivariate normal mean model where the mean vector is restricted to a closed convex cone with smooth or piecewise smooth boundary. We exemplify our estimators by two settings, one shrinking toward the ball and the other shrinking toward the cone of non-negative definite matrices.