中文摘要
本文研究处理分配问题中的一种惩罚统计决策规则。考虑一个功利主义政策制定者的设定，他必须使用样本数据，基于可观测特征将二元处理分配给总体中的成员。我们将此问题建模为一个统计决策问题，其中政策制定者必须从一类潜在子集中选择一个协变量空间的子集以分配处理。我们关注政策制定者可能希望在一组受约束的子集类中进行选择的情境：例子包括选择在其上进行最优子集选择的协变量数量，以及通过筛法逼近一个复杂类时的模型选择。我们借鉴并扩展统计学习中的结果，发展出惩罚福利最大化（PWM）规则。我们为PWM规则的遗憾建立了oracle不等式，表明它能够在可用类的集合上进行模型选择。随后我们利用这一oracle不等式推导出PWM最大遗憾的相关界。我们的结果的一个重要推论是，我们能够用“留出”程序将模型选择形式化，即政策制定者会先使用一半数据估计各种政策，然后选择在另一半数据上评估时表现最好的政策。
Abstract
This paper studies a penalized statistical decision rule for the treatment assignment problem. Consider the setting of a utilitarian policy maker who must use sample data to allocate a binary treatment to members of a population, based on their observable characteristics. We model this problem as a statistical decision problem where the policy maker must choose a subset of the covariate space to assign to treatment, out of a class of potential subsets. We focus on settings in which the policy maker may want to select amongst a collection of constrained subset classes: examples include choosing the number of covariates over which to perform best‐subset selection, and model selection when approximating a complicated class via a sieve. We adapt and extend results from statistical learning to develop the Penalized Welfare Maximization (PWM) rule. We establish an oracle inequality for the regret of the PWM rule which shows that it is able to perform model selection over the collection of available classes. We then use this oracle inequality to derive relevant bounds on maximum regret for PWM. An important consequence of our results is that we are able to formalize model‐selection using a “holdout” procedure, where the policy maker would first estimate various policies using half of the data, and then select the policy which performs the best when evaluated on the other half of the data.