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arXiv · 2607.13170

Nearly-minimax variance estimation under rough random design

P. M. Aronow·Patrick Lopatto
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Abstract

We identify the minimax exponent for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an $s$-H\"older regression function with $s>1$ in dimension $d>4s$, the minimax root-mean-square risk lies between $cn^{-2(s+1)/(d+4)}e^{-C\sqrt{\log n}}$ and $Cn^{-2(s+1)/(d+4)}$. These bounds show that the rate proposed by Robins is not uniformly attainable over this model class. For $0 1$ and $d\le4s$, it is $n^{-1/2}$.

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P. M. Aronow, Patrick Lopatto. 2026-07-14. Nearly-minimax variance estimation under rough random design. https://arxiv.org/abs/2607.13170

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