Empirical Bayes shrinkage methods usually maintain a prior independence assumption: The unknown parameters of interest are independent from the known precision of the estimates. This assumption is theoretically questionable and empirically rejected, and imposing it inappropriately may harm the performance of empirical Bayes methods. We instead model the conditional distribution of the parameter given the standard errors as a location-scale family, leading to a family of methods that we call CLOSE. We establish that (i) CLOSE is rate-optimal for squared error Bayes regret, (ii) squared error regret control is sufficient for an important class of economic decision problems, and (iii) CLOSE is worst-case robust. We use our method to select high-mobility Census tracts targeting a variety of economic mobility measures in the Opportunity Atlas (Chetty et al., 2020; Bergman et al., 2023). Census tracts selected by close are more mobile on average than those selected by the standard shrinkage method. For 6 out of 15 mobility measures considered, the gain of close over the standard shrinkage method is larger than the gain of the standard method over selecting Census tracts uniformly at random.

翻译：经验贝叶斯收缩方法通常维持一个先验独立性假设：感兴趣的未知参数与估计的已知精度相互独立。这一假设在理论上存疑且在实证中被拒绝，不恰当地施加该假设可能损害经验贝叶斯方法的性能。我们转而将给定标准误差的参数条件分布建模为位置-尺度族，由此产生一系列称为CLOSE的方法。我们证明了：(i) CLOSE在平方误差贝叶斯遗憾意义上达到率最优,(ii) 平方误差遗憾控制足以应对一类重要的经济决策问题,以及(iii) CLOSE具有最坏情况鲁棒性。我们运用该方法从机会图谱（Chetty等，2020；Bergman等，2023）中针对多种经济流动性指标筛选高流动性人口普查区。CLOSE筛选出的人口普查区平均流动性高于标准收缩方法筛选的样本。在15项流动性指标中，有6项指标显示CLOSE相较于标准收缩方法的增益大于标准方法相较于随机均匀选择人口普查区的增益。