The finite-population asymptotic theory provides a normal approximation for the sampling distribution of the average treatment effect estimator in stratified randomized experiments. The asymptotic variance is often estimated by a Neyman-type conservative variance estimator. However, the variance estimator can be overly conservative, and the asymptotic theory may fail in small samples. To solve these issues, we propose a sharp variance estimator for the difference-in-means estimator weighted by the proportion of stratum sizes in stratified randomized experiments. Furthermore, we propose two causal bootstrap procedures to more accurately approximate the sampling distribution of the weighted difference-in-means estimator. The first causal bootstrap procedure is based on rank-preserving imputation and we show that it has second-order refinement over normal approximation. The second causal bootstrap procedure is based on sharp null imputation and is applicable in paired experiments. Our analysis is randomization-based or design-based by conditioning on the potential outcomes, with treatment assignment being the sole source of randomness. Numerical studies and real data analyses demonstrate advantages of our proposed methods in finite samples.

翻译：有限总体渐近理论为分层随机试验中平均处理效应估计量的抽样分布提供了正态近似。其渐近方差通常采用内曼型保守方差估计量进行估计。然而，该方差估计量可能过于保守，且渐近理论在小样本情况下可能失效。针对这些问题，我们提出了一种针对分层随机试验中按层规模比例加权的均值差估计量的精确方差估计量。此外，我们提出了两种因果自助法，以更精确地逼近加权均值差估计量的抽样分布。第一种因果自助法基于秩保持插补法，我们证明其具有优于正态近似的二阶修正特性。第二种因果自助法基于尖锐零假设插补法，适用于配对试验。本研究以随机化为基础或设计为基础，通过以潜在结果为条件进行分析，仅将处理分配作为随机性来源。数值研究与实际数据分析证明了所提方法在有限样本中的优势。