Before revision:  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.
After revision:  The marginal optimization map is nonlinear and not differentiable, which complicates inference procedures, since statistical inference methods for nonlinear maps usually rely on compact differentiability. However, 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.