Before revision:  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:  We show that the map from objective function to uniform test statistics applied to the value function -- specifically, Kolmogorov-Smirnov or Cram\'er-von Mises statistics - are directionally differentiable.