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 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 shrinkage estimators towards Kj given by M. E. Bock (1982). In fact our estimators reduce to the James–Stein estimator and to the Bock estimator when K is a point and a convex 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 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 two settings, one shrinking to a ball and the other shrinking to the cone of nonnegative definite matrices.