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February 2012 A Geometrical Explanation of Stein Shrinkage
Lawrence D. Brown, Linda H. Zhao
Statist. Sci. 27(1): 24-30 (February 2012). 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.

Citation

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Lawrence D. Brown. Linda H. Zhao. "A Geometrical Explanation of Stein Shrinkage." Statist. Sci. 27 (1) 24 - 30, February 2012. https://doi.org/10.1214/11-STS382

Information

Published: February 2012
First available in Project Euclid: 14 March 2012

zbMATH: 1330.62282
MathSciNet: MR2953493
Digital Object Identifier: 10.1214/11-STS382

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.27 • No. 1 • February 2012
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