Summary: There is an innate human tendency, one might call it the “league table mentality,” to construct rankings. Schools, hospitals, sports teams, movies, and myriad other objects are ranked even though their inherent multi-dimensionality would suggest that – at best – only partial orderings were possible. We consider a large class of elementary ranking problems in which we observe noisy, scalar measurements of merit for \(n\) objects of potentially heterogeneous precision and are asked to select a group of the objects that are “most meritorious.” The problem is naturally formulated in the compound decision framework of H. Robbins’s [in: Proc. 3rd Berkeley Sympos. Math. Statist. Probability 1, 157–163 (1956; Zbl 0074.35302)] empirical Bayes theory, but it also exhibits close connections to the recent literature on multiple testing. The nonparametric maximum likelihood estimator for mixture models [J. Kiefer and J. Wolfowitz, Ann. Math. Stat. 27, 887–906 (1956; Zbl 0073.14701)] is employed to construct optimal ranking and selection rules. Performance of the rules is evaluated in simulations and an application to ranking U.S. kidney dialysis centers.
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