This paper develops a novel approach to uniform inference for functions that bound the distribution and quantile functions of heterogeneous treatment effects in randomized experiments when only marginal treatment and control distributions are observed and the joint distribution of outcomes is unobserved. These bounds are nonlinear maps of the marginal distribution functions of control and treatment outcomes, and  statistical inference methods for nonlinear maps usually rely on smoothness through a type of differentiability. We show that the maps from marginal distributions to bound functions are not differentiable, but uniform test statistics applied to the bound functions - such as Kolmogorov-Smirnov or Cram\'er-von Mises  - are directionally differentiable. We establish the consistency and weak convergence of nonparametric plug-in estimates of the test statistics and show how they can be used to conduct inference for bounds uniformly over the distribution of treatment effects. We also establish the directional differentiability of minimax operators applied to general - that is , not only convex-concave - functions, which may be of independent interest . In addition, we develop detailed resampling techniques to conduct practical inference for the bounds or for the true distribution or quantile function of the treatment effect distribution . Finally, we apply our methods to the evaluation of a job training program  .
This paper develops a novel approach to uniform inference for the optimal value function, that is, the function that results from optimizing an objective function marginally over one of its arguments. The marginal optimization map is nonlinear and not differentiable, which complicates inference procedures, since statistical inference methods for nonlinear maps usually rely on regularity through a type of differentiability. We show that the map from objective function to uniform test statistics applied to the value functions - such as Kolmogorov-Smirnov or Cram\'er-von Mises statistics - are directionally differentiable. We establish consistency and weak convergence of nonparametric plug-in estimates of the test statistics and show how they can be used to conduct uniform inference. Because the limiting distribution of sample value functions is not generally tractable, to conduct practical inference, we develop detailed resampling techniques that combine a bootstrap procedure with estimates of the directional derivatives . In addition, we formally establish uniform size control of the resampling procedure for testing. Monte Carlo simulations assess the finite-sample properties of the proposed methods and show accurate empirical size of the procedures . Finally, we apply our methods to the evaluation of a job training program using bounds for the distribution function of treatment effects .
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functions that bound the distribution and quantile functions of heterogeneous treatment effects in randomized experiments when only marginal treatment and control distributions are observed and the joint distribution of outcomes is unobserved. These bounds are nonlinear maps of the marginal distribution functions of control and treatment outcomes,
the optimal value function, that is, the function that results from optimizing an objective function marginally over one of its arguments. The marginal optimization map is nonlinear
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not differentiable, which complicates inference procedures, since
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smoothness
regularity
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maps from marginal distributions to bound functions are not differentiable, but
map from objective function to
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bound
value
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statistics
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the
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inference for bounds uniformly over the distribution of treatment effects. We also establish the directional differentiability of minimax operators applied to general - that is , not only convex-concave - functions, which may be of independent interest
uniform inference. Because the limiting distribution of sample value functions is not generally tractable, to conduct practical inference, we develop detailed resampling techniques that combine a bootstrap procedure with estimates of the directional derivatives
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develop detailed resampling techniques to conduct practical inference for the bounds or for the true distribution or quantile function of the treatment effect distribution
formally establish uniform size control of the resampling procedure for testing. Monte Carlo simulations assess the finite-sample properties of the proposed methods and show accurate empirical size of the procedures
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using bounds for the distribution function of treatment effects