A Geometrical Explanation of Stein Shrinkage
A Geometrical Explanation of Stein Shrinkage
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DOI：
10.1214/11-sts382
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发表时间：
2012-02
影响因子：
5.7
通讯作者：
L. Brown;Linda H. Zhao
中科院分区：
文献类型：
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作者：
L. Brown;Linda H. Zhao
收缩估计已经成为高维数据分析的基本工具。从历史上和概念上讲，这方面的一个关键发展是发现了多元正态平均值的通常估计量的不可接受性。本文对这一不允许性作了几何解释。通过利用球对称的问题，它是可能的，有效地概念化的多维设置在一个二维的框架，可以很容易地绘制和几何分析。我们开始从斯坦对不可接受性给出的启发式解释开始（见《第三届伯克利数理统计与概率研讨会论文集》，1954-1955年，第一卷（1956年），第197-206页，加州大学出版社）。一些几何图形包括使这个推理更有形。这也解释了为什么斯坦的论点福尔斯短产生一个证明的不可受理性，即使当尺寸，p，是远远大于p = 3。然后，我们扩展了几何思想，以产生越来越有说服力的论点，当p ≥ 3时，不容许的，虽然在增加的几何和计算细节的成本。
Shrinkage estimation has become a basic tool in the analysis of high-dimensional data. Historically and conceptually a key develop- ment 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 inadmissibil- ity. By exploiting the spherical symmetry of the problem it is possi- ble to effectively conceptualize the multidimensional setting in a two- dimensional framework that can be easily plotted and geometrically an- alyzed. 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.