A Geometrical Explanation of Stein Shrinkage

Loading...
Thumbnail Image
Penn collection
Statistics Papers
Degree type
Discipline
Subject
Stein estimation
shrinkage
minimax
empirical Bayes
high-dimensional geometry
Statistics and Probability
Funder
Grant number
License
Copyright date
Distributor
Related resources
Author
Brown, Lawrence D
Zhao, Linda H
Contributor
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.

Advisor
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
2012-01-01
Journal title
Statistical Science
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation
Collection