无设计光滑性的二元回归自适应置信集
Adaptive Confidence Sets for Binary Regression without Design Smoothness
AI总结：
该研究针对无设计光滑性的二元回归，构造了渐近诚实、速率最优的自适应置信集，回答了Mukherjee与Sen2018年提出的问题。
AI中文摘要：
我们在$L^2(dx)$损失下研究随机设计二元回归中回归函数的诚实自适应置信集。仅假设未知设计密度$g$满足已知界$0<c\leq g\leq C<\infty$，我们构造了渐近诚实、速率自适应的置信集，无需$g$是光滑的。当回归函数光滑度范围最多跨越两倍时，可实现完全自适应；在更宽的光滑度范围上，以对应测试速率$n^{-2s/(4s+d)}$在常规分离类上实现自适应。均匀设计下的下界表明这些分离速率是速率最优的，这回答了Mukherjee和Sen（2018）提出的问题。
英文摘要：
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).
