Statistics Papers

Document Type

Journal Article

Date of this Version

2012

Publication Source

Statistical Science

Volume

27

Issue

1

Start Page

24

Last Page

30

DOI

10.1214/11-STS382

Abstract

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 ≥ 3, albeit at the cost of increased geometric and computational detail.

Keywords

Stein estimation, shrinkage, minimax, empirical Bayes, high-dimensional geometry

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Date Posted: 27 November 2017

This document has been peer reviewed.