Memorial
Memorial
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纪念馆
DOI：
10.1177/0891988720917005
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发表时间：
2020
影响因子：
2.6
通讯作者：
Considerações Finais
中科院分区：
文献类型：
--
作者：
Considerações Finais
数理统计研究所 (IMS) 理事会批准了其纪念委员会的一项提案，将本期《统计年鉴》献给于 2016 年去世、享年 96 岁的查尔斯·M·斯坦因 (Charles M. Stein)。此次纪念活动反映了斯坦因作为一名数理统计学家的杰出成就，他的工作继续对该学科产生深远影响。作为联合编辑，我们征集了四篇文章，内容涉及斯坦因的一些最杰出和最持久的贡献。 Eaton 和 George（2021）描述了其中最早的一个，即 Stein 与 Gil Hunt 合作理解不变统计过程和极小极大之间的关系。人们已经知道，许多统计问题对于特定的变换组具有不变性，在这种情况下，很自然地要考虑尊重这种不变性的统计程序。正如伊顿和乔治所解释的那样，亨特和斯坦因着手了解所有程序的最小最大风险何时与不变程序的最小最大风险相匹配。它们不仅涵盖了亨特-斯坦定理所描述的优雅解决方案，还涵盖了丢失手稿的故事以及这些想法最终为人所知的方式。 Strawderman (2021) 讨论了 Stein 关于三个或更多维度多元正态均值的常用估计量不可接受的开创性结果 (Stein (1956b))。不变性在斯坦因的思想中也发挥了重要作用，因为对于所有球对称估计量的集合而言可接受的球对称估计量在所有估计量中也是可接受的。斯特劳德曼描述了 1956 年的论文如何成为对斯坦因收缩现象和（不）可采性进行进一步研究的启动台。不太直接的是，使用收缩来减少方差（以引入一些偏差为代价）的想法是岭回归和近年来变得如此流行的其他惩罚似然技术的核心。在著名的第三届伯克利数理统计与概率研讨会论文集的同一卷中，Stein (1956a) 为高效半参数推理理论奠定了基础，正如 van der Vaart 和 Wellner (2021) 所解释的那样。作者描述了斯坦因的见解，即当参数空间是无限维（非参数）时，可能存在一个一维最不利子问题，该子问题与估计基础参数的原始问题一样困难。这有助于将常规参数模型中的渐近估计理论扩展到无限维模型中的平滑泛函。在几个例子和扩展中，范德法特和韦尔纳提出了斯坦因估计由对称密度生成的位置族中的人口中位数的经典例子，表明缺乏这种对称密度的知识不会导致该中位数估计的渐近恶化。最后，Chen (2021) 概述了 Stein 的正态逼近方法，首先是他根据某些测试函数的期望消失来描述标准正态分布。事实证明，这产生了一种强大的技术，可以限制给定随机变量的分布与标准正态随机变量的分布之间的差异。陈提供了一些轶事来深入了解这一方式
The Institute of Mathematical Statistics (IMS) Council approved a proposal from its Committee on Memorials to dedicate this issue of the Annals of Statistics to Charles M. Stein, who died in 2016 aged 96. This memorialisation is a reflection of Stein’s distinction as a mathematical statistician, whose work continues to have a profound impact on the discipline. As co-editors, we have solicited four articles on some of the most remarkable and enduring of Stein’s contributions. Eaton and George (2021) describe the earliest of these, namely Stein’s work with Gil Hunt on understanding the relationship between invariant statistical procedures and minimaxity. It was already understood that many statistical problems possess invariance properties with respect to particular groups of transformations, and in such circumstances, it is natural to consider statistical procedures that respect this invariance. As Eaton and George explain, Hunt and Stein set out to understand when the minimax risk over all procedures matches the minimax risk over invariant ones. They cover not only the elegant solution as described by the Hunt–Stein theorem, but also the story of the lost manuscript and the way in which the ideas did finally come to be known. Strawderman (2021) discusses Stein’s seminal result on the inadmissibility of the usual estimator of a multivariate normal mean in three or more dimensions (Stein (1956b)). Invariance played a significant role in Stein’s thinking here too, in that a spherically symmetric estimator that is admissible with respect to the set of all spherically symmetric estimators turns out also to be admissible among all estimators. Strawderman describes the way in which the 1956 paper became a launchpad for many further investigations into the Stein shrinkage phenomenon and (in)admissibility. Less directly, the idea of using shrinkage to reduce variance (at the expense of introducing some bias) lies at the heart of ridge regression and other penalised likelihood techniques that have become so popular in recent years. Within the same volume of the famous Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Stein (1956a) laid the foundations for the theory of efficient semiparametric inference, as explained by van der Vaart and Wellner (2021). The authors describe Stein’s insight that when a parameter space is infinite-dimensional (nonparametric), there may be a one-dimensional least favourable subproblem that is as difficult as the original problem of estimating the underlying parameter. This facilitates an extension of the asymptotic theory of estimation in regular parametric models to one for smooth functionals in infinite-dimensional models. Among several examples and extensions, van der Vaart and Wellner present Stein’s classical example of estimating the population median in a location family generated by a symmetric density, showing that absence of knowledge of this symmetric density does not lead to an asymptotic deterioration in the estimation of this median. Lastly, Chen (2021) outlines Stein’s method for normal approximation, beginning with his characterisation of the standard normal distribution in terms of the vanishing of the expectations of certain test functions. This turns out to spawn a powerful technique for bounding a discrepancy between the distribution of a given random variable and that of a standard normal random variable. Chen includes several anecdotes that give an insight into the way