Improved Variance Estimation in Homoskedastic Nonparametric Random-Design Regression via a Two-Scale Approach

2026-09-08

AI Summary

研究通过两尺度方法改进非参数随机设计回归中的方差估计问题，针对不同平滑度条件提出新的估计器以提高估计精度。

Abstract

We study estimation of a constant conditional variance $\sigma^2$ 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 $\beta_b$-H\"older smooth, the design density is $\beta_g$-H\"older smooth and bounded above and away from zero, and we consider the nonparametric regime $\beta_b>1$ and $d>4\beta_b$. Set $\beta_g^\star=\beta_b(1-4\beta_b/d)/\{1+2\beta_b/d+8(\beta_b/d)^2\}$. We give an estimator whose mean squared error is upper bounded by $Cn^{-4(\beta_b+1)/(d+4)}$ in the low-regularity regime when $0<\beta_g\leq\beta_g^\star$. 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 $\beta_g>\beta_g^\star$, a higher-order influence function estimator of Robins, Li, Tchetgen Tchetgen, and van der Vaart (2008) provides the rate $Cn^{-8\beta_b/(d+4\beta_b)}$. 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.

Problem

• nonparametric regression
• random design
• variance estimation
• causal inference

Innovation

• Two-Scale Approach
• Variance Estimation
• Nonparametric Regression