Invidious Comparisons: Ranking and Selection as Compound Decisions

TL;DR

This paper develops optimal ranking and selection methods for objects with noisy, heterogeneous measurements, using empirical Bayes and mixture model techniques, and demonstrates their effectiveness through simulations and real-world application.

Contribution

It introduces a novel approach combining empirical Bayes and mixture models for ranking objects based on noisy data, addressing multi-dimensionality and heterogeneity.

Findings

The proposed methods outperform traditional ranking procedures in simulations.

Application to U.S. kidney dialysis centers shows practical effectiveness.

The approach provides a statistically optimal framework for compound decision problems.

Abstract

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 Robbins's (1956) empirical Bayes theory, but it also exhibits close connections to the recent literature on multiple testing. The nonparametric maximum likelihood estimator for mixture models (Kiefer and Wolfowitz (1956)) 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.