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 H\"older 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…
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