- Statist. Sci.
- Volume 27, Number 1 (2012), 24-30.
A Geometrical Explanation of Stein Shrinkage
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.
Statist. Sci., Volume 27, Number 1 (2012), 24-30.
First available in Project Euclid: 14 March 2012
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Brown, Lawrence D.; Zhao, Linda H. A Geometrical Explanation of Stein Shrinkage. Statist. Sci. 27 (2012), no. 1, 24--30. doi:10.1214/11-STS382. https://projecteuclid.org/euclid.ss/1331729980