Sharp bounds on the distribution of treatment effects and their statistical inference

In this paper, we propose nonparametric estimators of sharp bounds on the distribution of treatment effects of a binary treatment and establish their asymptotic distributions. We note the possible failure of the standard bootstrap with the same sample size and apply the fewer-than-n bootstrap to making inferences on these bounds. The finite sample performances of the confidence intervals for the bounds based on normal critical values, the standard bootstrap, and the fewer-than-n bootstrap are investigated via a simulation study. Finally we establish sharp bounds on the treatment effect distribution when covariates are available.