We study the problem of selecting the best $m$ units from a set of $n$ as $m / n \to \alpha \in (0, 1)$, where noisy, heteroskedastic measurements of the units' true values are available and the decision-maker wishes to maximize the average true value of the units selected. Given a parametric prior distribution, the empirical Bayes decision rule incurs $O_p(n^{-1})$ regret relative to the Bayesian oracle that knows the true prior. More generally, if the error in the estimated prior is of order $O_p(r_n)$, regret is $O_p(r_n^2)$. In this sense selecting the best units is easier than estimating their values. We show this regret bound is sharp in the parametric case, by giving an example in which it is attained. Using priors calibrated from a dataset of over four thousand internet experiments, we find that empirical Bayes methods perform well in practice for detecting the best treatments given only a modest number of experiments.

翻译：我们研究从一组美元(0,1,1美元)中选择最优单位美元的问题,从一组美元中选择最优单位美元($m / n \ to pha \ in (0, 1, 1美元),在那里,可以对单位的真实值进行吵闹的、 hestrokedatic 的测量,而决策者希望使所选单位的平均真实值最大化。鉴于先前分布的参数,经验性贝耶斯决定规则对了解以前真实情况的巴耶斯人或古灵来说,产生0美元(n)-1美元(美元)的遗憾。更一般地说,如果之前估计的错误是美元($_p(r_n),则遗憾是$O_p(r_n)2美元)。从这个意义上讲,选择最佳单位比估计其价值容易得多。我们通过举例来表明,在参数性案例中,这种遗憾的界限是尖锐的。我们发现,根据四千多个互联网实验的数据集校准,经验性贝耶斯方法在实际中在检测只给少量实验的最佳处理方法方面表现良好。