发表机构
University of Pennsylvania; Harvard T.H. Chan School of Public Health（宾夕法尼亚大学; 哈佛大学陈曾熙公共卫生学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维非参数回归中的常数方差估计，提出两尺度方法，在低正则性下达到最优均方误差速率，并给出高正则性下的高阶影响函数估计及全对岭扩展，模拟验证了有效性。
AI 中文摘要
我们研究了在具有d维随机设计的非参数回归中估计常数条件方差σ²的问题。这是一个重要的问题，类似的疑问也出现在因果推断中。回归函数是β_b-Hölder光滑的，设计密度是β_g-Hölder光滑的，并且有上界和下界（远离零），我们考虑非参数情形β_b>1且d>4β_b。设β_g^⋆=β_b(1-4β_b/d)/{1+2β_b/d+8(β_b/d)²}。我们给出了一个估计量，在低正则性情形（0<β_g≤β_g^⋆）下，其均方误差的上界为Cn^{-4(β_b+1)/(d+4)}。低正则性分支基于一种新的两尺度构造：将协变量空间划分为单元，在每个合适的单元内投影出局部多项式趋势，并将每个单元中一对合格近邻的平方归一化对比度在单元间取平均。在高正则性情形（β_g>β_g^⋆）下，Robins、Li、Tchetgen Tchetgen和van der Vaart（2008）的高阶影响函数估计量提供了速率Cn^{-8β_b/(d+4β_b)}。我们还给出了一个全对岭扩展，它达到了相同的两尺度速率，并在模拟中将该方法与一系列现有估计量进行了评估。
英文摘要
We study estimation of a constant conditional variance $σ^2$ in nonparametric regression with a $d$-dimensional random design. This is an important problem, and similar questions arise in causal inference. The regression function is $β_b$-Hölder smooth, the design density is $β_g$-Hölder smooth and bounded above and away from zero, and we consider the nonparametric regime $β_b>1$ and $d>4β_b$. Set $β_g^\star=β_b(1-4β_b/d)/\{1+2β_b/d+8(β_b/d)^2\}$. We give an estimator whose mean squared error is upper bounded by $Cn^{-4(β_b+1)/(d+4)}$ in the low-regularity regime when $0<β_g\leqβ_g^\star$. The low-regularity branch is based on a new two-scale construction: the covariate space is partitioned into cells, the local polynomial trend is projected out within each suitable cell, and the squared normalized contrast from one eligible close pair per cell is averaged across cells. In the high regularity regime when $β_g>β_g^\star$, a higher-order influence function estimator of Robins, Li, Tchetgen Tchetgen, and van der Vaart (2008) provides the rate $Cn^{-8β_b/(d+4β_b)}$. We also give an all-pairs ridge extension, which achieves the same two-scale rate, and evaluate the methods alongside a range of existing estimators in simulations.
