Estimating quantile treatments without strict overlap

We consider the problem of estimating quantile treatment effects without assuming strict overlap , i.e., we do not assume that the propensity score is bounded away from zero. More specifically, we consider an inverse probability weighting (IPW) approach for estimating quantiles in the potential outcomes framework and pay special attention to scenarios where the propensity scores can tend to zero as a regularly varying function. Our approach effectively considers a heavy-tailed objective function for estimating the quantile process. We introduce a truncated IPW estimator that is shown to outperform the standard quantile IPW estimator when strict overlap does not hold. We show that the limiting distribution of the estimated quantile process follows an infinitely divisible law and converges at the rate $n^{1-1/γ}$, where $γ>1$ is the tail index of the propensity scores when they tend to zero. We propose a practical, data-driven procedure for selecting the truncation parameter, grounded in our asymptotic theory. The performance of our estimators is illustrated in numerical experiments and in a dataset that exhibits the presence of extreme propensity scores.