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\hypersetup{pdftitle={Nearly Minimax Rates for Functional Estimation Under Rough Random Design},pdfauthor={P. M. Aronow, Nathan Kallus, and Patrick Lopatto}}
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% Short running heads for section titles that do not fit beside their chapter title.
\shorthead{Embedding the priors in the generic model}{Embedding the priors}
\shorthead{Proof of \texorpdfstring{\cref{cor:applications}}{the corollary} for these targets}{Proof for these targets}
\begin{document}
\frontmatter
\npptitlepage{Nearly Minimax Rates\\ for Functional Estimation\\ Under Rough Random Design}{P.\ M.\ Aronow\\ Nathan Kallus\\ Patrick Lopatto}
\begin{authorsnote}
This paper was written by the authors with the assistance of large language
models, which were used to suggest mathematical arguments, draft and revise
exposition, and write code for computational verification. We have taken care
to supervise this writing to ensure that the proofs meet a minimum standard of
readability and that previous literature is appropriately cited. However, the
resulting document remains lacking in exposition and overall coherence. We are
nonetheless releasing this manuscript on Hexagon to disseminate the result as
quickly as possible, because we believe it may be of interest to the statistics
community. We welcome suggestions concerning mathematical corrections, omitted
citations, and attribution of ideas.

P.L.\ was partially supported by NSF grant DMS-2450004.
\end{authorsnote}
\begin{abstractpage}
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 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.
\end{abstractpage}
\tableofcontents
\mainmatter
\input{body}
\backmatter
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\end{document}