A Geometrical Explanation of Stein Shrinkage

TLDR: In this article, a geometrical explanation for the inadmissibility of the usual estimator of a multivariate normal mean is presented, which is based on the spherical symmetry of the problem.

Abstract: Shrinkage estimation has become a basic tool in the analysis of high-dimensional data. Historically and conceptually a key develop- ment 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 inadmissibil- ity. By exploiting the spherical symmetry of the problem it is possi- ble to effectively conceptualize the multidimensional setting in a two- dimensional framework that can be easily plotted and geometrically an- alyzed. 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 ≥ 3, albeit at the cost of increased geometric and computational detail.