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Nearly Minimax Rates for Functional Estimation Under Rough Random Design

Primarily AI-generated textHuman understanding: some partsmath.ST — Statistics Theory

Contributed by P. M. Aronow ↗, Nathan Kallus ↗, Patrick Lopatto ↗

Version 1 / Oct 08, 2026 / CC BY 4.0

Abstract

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 ss in dimension dd, the minimax root-mean-square error is n−2s/d+o(1)n^{-2s/d+o(1)} when s<d/4s<d/4 and of order n−1/2n^{-1/2} when s≥d/4s\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−κlog⁡ne^{-\kappa\sqrt{\log n}} when s<d/4s<d/4, with κ\kappa explicit in terms of s/ds/d and the weight bounds. For the expected conditional covariance, the same exponent and the same constant κ\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(log⁡n)o(\sqrt{\log n}) remainder in the logarithm of the risk.

Provenance statement

Proof discovery and verification were done with a combination of GPT 5.6 Sol, GPT 6 Sol, GPT 6 Astra, GPT 6.1 Sol, Claude Fable 5 and 5.1, and Claude Opus 5.5. Lean formalizations were produced by GPT 6.1 Sol as part of verification. An early version of the Abstract and Introduction was drafted by humans, and earlier version of the proofs had edits performed by humans for clarity. The final version, involving stronger results, was substantially redrafted by Opus 5.5 and GPT 6 Astra.

Formalizations

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude FableVersion 5.1
Anthropic
Claude FableVersion 5
OpenAI
ChatGPTVersion 6.1
OpenAI
ChatGPTVersion 6
OpenAI
ChatGPTVersion 5.6
Lean
LeanVersion 4.34.1

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Version history

  1. v1Submitted by P. M. AronowInitial depositCurrentOct 08, 2026