General Forms of Finite Population Central Limit Theorems with Applications to Causal Inference
General Forms of Finite Population Central Limit Theorems with Applications to Causal Inference
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DOI：
10.1080/01621459.2017.1295865
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发表时间：
2016-10
影响因子：
3.7
通讯作者：
Xinran Li;Peng Ding
中科院分区：
文献类型：
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作者：
Xinran Li;Peng Ding
频率主义者的推理通常提供与感兴趣的参数的置信区间或集合相关联的点估计。构造置信区间或置信集需要理解点估计量的抽样分布，在许多但并非所有情况下，这些分布与由中心极限定理确保的渐近正态分布有关。虽然以前的文献已经建立了各种形式的中心极限定理的统计推断在超人口模型，我们仍然需要一些随机化为基础的因果分析的实验数据，其中的参数是有限的人口和随机性的函数仅来自于治疗分配的一般和方便的形式的中心极限定理。利用样本调查和秩统计的中心极限定理，建立了有限总体中心极限定理的一般形式，该定理对于证明零个体因果效应的尖锐零假设下随机化检验的渐近分布，以及获得因果效应估计量的渐近重复抽样分布特别有用.新的中心极限定理适用于多处理水平、多处理因子和向量结果的一般试验设计，并可直接应用于研究因果推断中的许多方法的渐近性质，包括工具变量、回归调整、重新随机化、聚类随机化试验等。它实际上依赖于以前没有建立的有限总体中心极限定理的一般形式。我们的新定理填补了这一空白，为基于渐近随机化的因果推理提供了更坚实的理论基础。本文的补充材料可在网上查阅。
ABSTRACT Frequentists’ inference often delivers point estimators associated with confidence intervals or sets for parameters of interest. Constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analyses of experimental data, where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal effects, and for obtaining the asymptotic repeated sampling distributions of the causal effect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels, multiple treatment factors and vector outcomes, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, cluster-randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of finite population central limit theorems that have not been established before. Our new theorems fill this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference. Supplementary materials for this article are available online.