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On Rereading Stein’s Lemma: Its Intrinsic Connection with Cramér-Rao Identity and Some New Identities
Methodology and Computing in Applied Probability ( IF 0.9 ) Pub Date : 2020-10-31 , DOI: 10.1007/s11009-020-09830-w
Nitis Mukhopadhyay

Now is an opportune time to revisit Stein’s (1973) beautiful lemma all over again. It is especially so since researchers have recently begun discovering a great deal of potential of closely related Stein’s unbiased risk estimate (SURE) in a number of directions involving novel applications. In recognition of the importance of the topic of Stein’s (1973; Ann Statist 9:1135–1151, 1981) research and its elegance, we include a selective review from the field. The process of rereading Stein’s lemma and reliving its awesome simplicity as well as versatility rekindled a number of personal thoughts, queries, and observations. A number of new and interesting insights are highlighted in the spirit of providing updated and futuristic versions of the celebrated lemma by largely focusing on univariate continuous distributions not belonging to an exponential family. In doing so, a number of new identities have emerged when the parent population is continuous, but they are highly non-normal. Last, but not the least, we have argued that there is no big foundational difference between the basic messages obtained via Stein’s identity and Cramér-Rao identity.



中文翻译:

重读斯坦因引理:它与克拉默罗身份和一些新身份的内在联系

现在是重新审视斯坦(1973)的美丽引理的时候了。特别是因为研究人员最近开始发现与斯坦因无偏风险估计密切相关的巨大潜力。(SURE)在涉及新颖应用的许多方向上。认识到斯坦因(1973; Ann Statist 9:1135–1151,1981)研究主题的重要性及其高雅性,我们对这一领域进行了选择性回顾。重读斯坦因引理并恢复其出色的简单性和多功能性的过程重新燃起了许多个人的想法,疑问和观察。本着通过提供单变量连续分布而不是单变量连续分布来提供著名引理的更新和未来版本的精神,强调了许多新的有趣的见解。属于一个指数家族。这样做时,当父母群体连续时,出现了许多新的身份,但它们是高度不正常的。最后但并非最不重要的一点是,我们认为,通过斯坦因的身份获得的基本信息与克拉默尔-饶的身份所获得的基本信息之间并没有太大的根本区别。

更新日期:2020-11-02
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