发表机构
Yonsei University（延世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对协变量密度仅知上下界时，提出多分辨率估计器以最小最大意义估计期望条件协方差，证明最优速率优于先前猜想并解决长期猜想，实验显示其均方误差更小。
AI 中文摘要
在许多科学研究中，评估两个响应变量之间的关联需要调整共享协变量。期望条件协方差衡量这种调整后的关联，并用于条件独立性检验和处理效应估计。研究者通常使用拟合的条件均值来估计它，但消除由这些条件均值的估计误差引起的偏差仍然是一项困难的任务。当真协变量密度未知且仅受已知上下界限制时，这一困难可能会增加。在本文中，我们研究在这些密度界下期望条件协方差能被估计的准确程度。我们提出了一种多分辨率估计器，在不估计密度的情况下减少偏差。理论结果表明，对于有界响应和粗糙的回归函数，与已知密度相比，未知密度会使最优最坏情况速率减慢一个样本量的幂次。我们证明，仅密度界下的最优速率严格优于先前推测的多项式速率，改进了一个次多项式因子，解决了一个长期存在的猜想。我们的估计器达到该速率，直至对数因子。通过模拟和对钻石数据的真实数据重采样实验，我们表明我们的方法可以比标准未校正估计器产生更小的均方误差。
英文摘要
In many scientific studies, evaluating the association between two responses requires adjusting for shared covariates. The expected conditional covariance measures this adjusted association and is used in conditional independence testing and treatment effect estimation. Researchers often estimate it using fitted conditional means, but removing the bias caused by the estimation errors of these conditional means remains a difficult task. This difficulty can increase when the true covariate density is unknown and restricted only by known upper and lower bounds. In this article, we study how accurately the expected conditional covariance can be estimated under these density bounds. We propose a multiresolution estimator that reduces bias without estimating the density. Theoretical results show that, for bounded responses and rough regression functions, an unknown density slows the optimal worst-case rate by a power of sample size compared with a known density. We establish that the optimal rate under density bounds alone strictly improves upon the previously conjectured polynomial rate by a sub-polynomial factor, resolving a long-standing conjecture. Our estimator attains this rate up to a logarithmic factor. Through simulations and a real data resampling experiment on diamonds, we show that our method can yield smaller mean squared errors than standard uncorrected estimators.
Comments27pages, 2 figures
