Sharp variance estimator and causal bootstrap in stratified randomized experiments

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.