On a Problem of Robbins
نویسندگان
چکیده
Robbins’s visionary 1951 paper can be seen as an exercise in binary classification, but also as a precursor to the outpouring of recent work on high-dimensional data analysis and multiple testing. It can also be seen as the birth of empirical Bayes methods. Our objective in the present note is to use this problem and several variants of it to provide a glimpse into the evolution of empirical Bayes methods. Much more comprehensive surveys of empirical Bayes methods and their modern relevance are provided by Zhang (2003) and Efron (2010); here, we aspire only to tell a more condensed version of the story, but one that highlights the critical role that the nonparametric maximum likelihood estimator (NPMLE) of Kiefer & Wolfowitz (1956) can play. Recent developments in convex optimization, as argued in Koenker & Mizera (2014), have greatly expanded the applicability of the Kiefer–Wolfowitz estimator and thereby increased the potential scope of nonparametric empirical Bayes methods. In prior work, Koenker & Mizera (2014), Koenker (2014), Koenker & Gu (2013), and Gu & Koenker (2014), we have emphasized the role of the Kiefer–Wolfowitz NPMLE in various estimation problems typically under squared-error loss. In this paper, in contrast, we will stress its potential usefulness mainly in classification and multiple testing.
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