The Stein paradox has played an influential role in the field of high-dimensional statistics. This result warns that the sample mean, classically regarded as the “usual estimator,” may be suboptimal in high dimensions. The development of the James–Stein estimator that addresses this paradox has by now inspired a large literature on the theme of “shrinkage” in statistics. In this direction, we develop a James–Stein-type estimator for the first principal component of a high-dimension and low-sample size data set. This estimator shrinks the usual estimator, an eigenvector of a sample covariance matrix under a spiked covariance model, and yields superior asymptotic guarantees. Our derivation draws a close connection to the original James–Stein formula so that the motivation and recipe for shrinkage is intuited in a natural way.

中文翻译：

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第一主成分的 James-Stein 估计

斯坦因悖论在高维统计领域发挥了重要作用。该结果警告说，经典地被视为“通常的估计量”的样本均值在高维度上可能不是最优的。解决这一悖论的 James-Stein 估计器的发展现在已经激发了关于统计学中“收缩”主题的大量文献。在这个方向上，我们为高维和低样本量数据集的第一个主成分开发了一个 James-Stein 型估计器。该估计量缩小了通常的估计量，即尖峰协方差模型下样本协方差矩阵的特征向量，并产生了优越的渐近保证。我们的推导与原始的 James-Stein 公式密切相关，因此可以以自然的方式直观地了解收缩的动机和配方。