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In this paper the estimation of the distribution function for potentialoutcomes to receiving or not receiving a treatment is studied. The approach isbased on weighting observed data on the basis on estimated propensity score. Aweighted version of empirical process is constructed and its weak convergenceto bivariate Gaussian process is established. Results for the estimation of theAverage Treatment Effect (ATE) and Quantile Treatment Effect (QTE) are obtainedas by-products. Applications to the construction of nonparametric tests for thetreatment effect and for the stochastic dominance of the treatment over controlare considered, and their finite sample properties and merits are studied viasimulation.

Estimation of distributional effects of treatment and control under selection on observables: consistency, weak convergence, and applications