with the treatment effect Δ = Y$_{1}$ − Y$_{0}$ where Y$_{0}$ and Y$_{1}$ denote the potential outcomes without and with some treatment, respectively. The question that I am interested in is how treatment effects or program benefits are distributed across the population. In the context of welfare policy evaluation, distributional aspects of the effects are often of interest, e.g. ”which individuals are severely affected by the program?” or ”how are those benefits distributed across the population?”. As Heckman et al. (1997) pointed out, the DTE is particularly important when treat- ments produce nontransferable and nonredistributable benefits such as outcomes in health interventions, academic achievement in educational programs, and occupational skills in job training programs or when some individuals experience severe welfare changes at the tails of the impact distribution. Although most empirical research on program evaluation has focused on average treatment effects (ATE) or marginal distributions of potential outcomes, these parameters are limited in their ability to capture heterogeneity of the treatment effects at the individual level. For example, consider two projects with the same average benefits, one of which concentrates benefits among a small group of people, while the other distributes benefits evenly across the population. ATE cannot differentiate between the two projects because it shows only the central tendency of treatment effects as a location parameter, whereas the DTE captures information about the entire distribution. Marginal distributions of Y$_{0}$ and Y$_{1}$ are also uninformative about parameters on the individual specific heterogeneity in treatment effects including the fraction of the population that benefits from a program Pr ( Y$_{1}$ ≥ Y$_{0}$ ) , the fraction of the population that has gains or losses in a specific range Pr $^{(}$δ L ≤ Y$_{1}$ − Y$_{0}$ ≤ δ U $^{)}$, the q -quantile of the impact distribution inf { δ : F$_{Δ}$ ( δ ) > q } , etc. See, e.g. Heckman et al. (1997), Abbring and Heckman (2007), and Firpo and Ridder (2008), among others for more details. Despite the importance of these parameters in economics, related empirical research has been hampered by difficulties associated with identifying the entire distribution of effects. The central challenge arises from a missing data problem: under mutually exclusive treatment participation, econometricians can observe either a treated outcome or an untreated outcome, but both potential outcomes Y$_{0}$ and Y$_{1}$ are never simultaneously observed for each agent. Therefore, the joint distribution of Y$_{0}$ and Y$_{1}$ is not typically exactly identified, which complicates identification of the DTE, which is point-identified only under strong assumptions about each individual’s rank across the treatment status or specifications on the joint distribution of Y$_{0}$ and Y$_{1}$ , which are often not justified by economic theory or plausible priors. This paper relies on partial identification to avoid strong assumptions and remain cautious of assumption- driven conclusions. In the related literature, Manski (1997) established bounds on the DTE under monotone treatment response (MTR), which assumes that the treatment effects are nonnegative. Fan and Park (2009, 2010) and Fan and Wu (2010) adopted results from copula theory to establish bounds on the DTE, given 2