Empirical Bayes methods usually maintain a prior independence assumption: The unknown parameters of interest are independent from the known standard errors of the estimates. This assumption is often theoretically questionable and empirically rejected. This paper instead models the conditional distribution of the parameter given the standard errors as a flexibly parametrized family of distributions, leading to a family of methods that we call CLOSE. This paper establishes 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 when our assumption on the conditional distribution is misspecified. Empirically, using CLOSE leads to sizable gains for selecting high-mobility Census tracts. Census tracts selected by CLOSE are substantially more mobile on average than those selected by the standard shrinkage method.

翻译：经验贝叶斯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标准误差相互独立。这一假设在理论上常存疑，并在实证中遭到拒绝。本文转而将给定标准误差下的参数条件分布建模为一个灵活参数化的分布族，由此衍生出一系列我们称为CLOSE的方法。本文证明了：(i) CLOSE在平方误差贝叶斯后悔值上达到速率最优；(ii) 对于一类重要的经济决策问题，平方误差后悔控制是充分的；(iii) 当我们的条件分布假设被错误设定时，CLOSE具有最坏情况下的稳健性。实证中，使用CLOSE能在筛选高流动性人口普查区时带来显著收益。与标准压缩方法选出的区域相比，CLOSE选出的人口普查区平均流动性明显更高。