Uniform-design validity of the Kennedy–2024 minimax lower bound

Determine whether the minimax lower bound established by Kennedy (2024) for pointwise conditional average treatment effect estimation under a design density that vanishes on parts of the covariate domain also holds when the covariate design density is uniform.

Background

The paper studies minimax estimation of the conditional average treatment effect at a fixed covariate point. Prior work by Kennedy (2024) established minimax lower bounds and constructed a higher-order local-polynomial R-learner under suitable design-density conditions, but the lower-bound construction used a density that vanishes on parts of the covariate domain.

The unresolved issue is whether that same lower-bound rate remains valid for a uniform covariate design. The paper resolves this question in a restricted regime—interior covariate points with low nuisance smoothness—but the quoted passage explicitly identifies the uniform-design extension as an open problem at that point in the introduction.