Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary
Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary
复制标题
具有平滑边界的闭凸集的收缩估计
DOI：
10.1006/jmva.1999.1895
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
A. Takemura
中科院分区：
文献类型：
--
作者：
S. Kuriki;A. Takemura
我们给詹姆斯？多元正态均值向量的Stein型估计，对光滑或分段光滑边界的闭凸集K收缩。收缩率由K在K上的投影点处的边界曲率决定。通过考虑一个序列的多面体Kj收敛到K，我们表明，我们提出的一个特定的估计是一个序列的收缩估计Kj的极限由M。E. Bock（1982）.事实上，我们的估计减少到詹姆斯？Stein估计和Bock估计。因此，它们可以被认为是这些估计量的自然扩展。此外，我们还将同样的方法应用于多元正态平均模型中的约束模型，即平均向量被限制在一个具有光滑或分段光滑边界的闭凸锥上的模型。我们证明了我们的估计在两个设置，一个收缩到一个球，另一个收缩到锥的非负定矩阵。
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.