Univariate Nonparametric Regression in the Presence of Auxiliary Covariates
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存在辅助协变量时的一元非参数回归
- 研究问题
- 当辅助协变量的作用形成较大回归噪声、掩盖目标预测变量与响应的关系时，如何恢复可靠的一元非参数回归估计？
- 方法与思路
- 作者建模并减少辅助协变量产生的噪声，建立以已知辅助效应的 oracle 为基准的渐近理论，并提出小样本去噪散点图程序。
- 材料陈述的贡献
- 摘要称，方法可在渐近意义下达到 oracle 表现；模拟及含八十四个观测、九个辅助协变量的数据例子支持其实用可行性。
- 阅读关注点
- 应核对 oracle 最优性的损失与模型条件、辅助效应的估计误差，以及小样本程序、形状限制和可视化解释的有效范围。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；具体最优性条件与误差数值未说明，辅助变量调整也不能自动赋予回归关系因果含义。
原文摘要
This article addresses the problem of finding a relationship between the univariate predictor and the response when regression errors, created in part by known auxiliary covariates, are too large for a reliable regression estimation. A typical example is a controlled random design experiment with a large number of covariates, where the statistician is interested in the effect of a particular covariate and this effect is blurred by a large regression noise created by other covariates. This article develops a theory of asymptotically optimal nonparametric univariate regression estimation in the presence of auxiliary covariates. Here optimality means mimicking the performance of an oracle that knows the effects of auxiliary covariates on the response. The asymptotic theory shows that such an optimal estimation is possible, and also explains how to evaluate the noise created by auxiliary covariates and how to develop an estimator for the interesting case of small sample sizes. The concept of modeling regression noise is well known in analysis of covariance (ANCOVA), and here it is applied in the optimal way to a nonparametric regression setting. A procedure for small sample sizes, denoised scattergram, is tested on simulated examples and a real dataset with 84 observations and 9 auxiliary covariates; the results justify the practical feasibility of the developed method. The method also allows a practitioner to visualize how a dataset would appear if the effects of auxiliary covariates were eliminated and to determine why an exhibited regression function has any given particular shape. Many practical recommendations (in particular, how to use known shape restrictions) are presented and discussed. The asymptotic theory, a numerical study, and analysis of a real dataset indicate that the proposed method of reducing the variance of regression errors created by auxiliary covariances is feasible, is easy to implement, and improves the likelihood of a meaningful regression analysis.