Adaptive Confidence Sets for Binary Regression without Design Smoothness

P. M. Aronow; Patrick Lopatto

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arXiv · 2608.01309

Adaptive Confidence Sets for Binary Regression without Design Smoothness

P. M. Aronow·Patrick Lopatto

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Abstract

We study honest adaptive confidence sets for the regression function in random-design binary regression under $L^2(dx)$ loss. Assuming only known bounds $0<c\leq g\leq C<\infty$ on the unknown design density, we construct asymptotically honest, rate-adaptive confidence sets without requiring $g$ to be smooth. Full adaptation is possible when the range of regression-function smoothness spans at most a factor of two. Over wider smoothness ranges, adaptation is achieved on the usual separated classes at the corresponding testing rates $n^{-2s/(4s+d)}$. A lower bound under the uniform design shows that these separation rates are rate-optimal. This answers a question raised by Mukherjee and Sen (2018).

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P. M. Aronow, Patrick Lopatto. 2026-08-02. Adaptive Confidence Sets for Binary Regression without Design Smoothness. https://arxiv.org/abs/2608.01309

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