Statistics – Methodology
Scientific paper
2012-03-21
Statistical Science 2012, Vol. 27, No. 1, 24-30
Statistics
Methodology
Published in at http://dx.doi.org/10.1214/11-STS382 the Statistical Science (http://www.imstat.org/sts/) by the Institute of M
Scientific paper
10.1214/11-STS382
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.
Brown Lawrence D.
Zhao Linda H.
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