A Geometrical Explanation of Stein Shrinkage

• Published in: arXiv.org 2012-03
• Main Authors: Brown, Lawrence D, Zhao, Linda H
• Format: Article
• Language: English
• Subjects: Dimensional analysis; Shrinkage
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1203.4737

Shrinkage estimation has become a basic tool in the analysis of high-dimensional data. Historically and conceptually a key development toward this was the discovery of the inadmissibility of the usual estimator of a multivariate normal mean. This article develops a geometrical explanation for this inadmissibility. By exploiting the spherical symmetry of the problem it is possible to effectively conceptualize the multidimensional setting in a two-dimensional framework that can be easily plotted and geometrically analyzed. We begin with the heuristic explanation for inadmissibility that was given by Stein [In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954--1955, Vol. I (1956) 197--206, Univ. California Press]. Some geometric figures are included to make this reasoning more tangible. It is also explained why Stein's argument falls short of yielding a proof of inadmissibility, even when the dimension, \(p\), is much larger than \(p=3\). We then extend the geometric idea to yield increasingly persuasive arguments for inadmissibility when \(p\geq3\), albeit at the cost of increased geometric and computational detail.