Nearly Minimax Rates for Functional Estimation Under Rough Random Design

We establish nearly minimax bounds for missing-at-random means, treatment effects, and expected conditional covariances under rough random design. For two nuisance functions with average Hölder smoothness $s$ in dimension $d$, the minimax root-mean-square error is $n^{-2s/d+o(1)}$ when $s<d/4$ and of order $n^{-1/2}$ when $s\ge d/4$. The first rate confirms the rough-design exponent suggested by higher-order influence function theory. In the generic model with an unknown bounded density weight, our upper and lower bounds differ by only polylogarithmic factors and identify a leading correction $e^{-\kappa\sqrt{\log n}}$ when $s<d/4$, with $\kappa$ explicit in terms of $s/d$ and the weight bounds. For the expected conditional covariance, the same exponent and the same constant $\kappa$ were obtained independently and concurrently by S. Park (arXiv:2610.05006). For models with separately bounded density and propensity, we identify the same polynomial exponent and the explicit leading correction, with an $o(\sqrt{\log n})$ remainder in the logarithm of the risk.