Density and treatment effect estimation under covariate-adaptive randomization with heavy-tailed outcomes.

Randomized experiments are the gold standard for investigating causal relationships, with comparisons of potential outcomes under different treatment groups used to estimate treatment effects. However, outcomes with heavy-tailed distributions pose significant challenges to traditional causal inference approaches. While recent studies have explored these issues under simple randomization, their application in more complex randomization designs, such as stratified randomization or covariate-adaptive randomization, has not been adequately addressed. To fill the gap, we first investigate the performance of nonparametric kernel density estimation methods under covariate-adaptive randomization, thereby establishing theoretical guarantees for treatment effect estimators based on the estimated densities. Second, we demonstrate the application of our density estimation framework to estimate the density treatment effect and overall quantile treatment effect, deriving the consistency and asymptotic normality of the estimators. For the overall quantile treatment effect, we show that the existing variance estimator for the influence function-based M-estimator tends to overestimate the asymptotic variance, especially under more balanced designs, and lacks universal applicability across randomization methods. To remedy this, we introduce a novel stratified transformed difference-in-means estimator to enhance efficiency and propose a universally applicable variance estimator to facilitate valid inferences. Numerical results demonstrate the effectiveness of the proposed methods in finite samples.