Design-adaptive Nonparametric Regression
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适应设计特征的非参数回归
- 研究问题
- 如何构造能适应随机或固定设计、聚集或均匀设计以及内部或边界位置的非参数回归平滑方法，并评估其效率？
- 方法与思路
- 作者研究加权局部线性回归，并在包括核、正交级数和样条方法的线性平滑器范围内分析渐近效率，辅以模拟。
- 材料陈述的贡献
- 摘要称，局部线性平滑器具有设计适应性，无须另作边界修正；适当选择核与带宽时，所述渐近效率可达到百分之百。
- 阅读关注点
- 应核对效率的定义及比较范围、达到效率界所需的核和带宽条件，以及高度聚集设计下的假设与模拟设置。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；效率陈述依赖渐近框架与调参条件，不能解释为所有样本中均最优，也不能推断刊本变化。
原文摘要
In this article we study the method of nonparametric regression based on a weighted local linear regression. This method has advantages over other popular kernel methods. Moreover, such a regression procedure has the ability of design adaptation: It adapts to both random and fixed designs, to both highly clustered and nearly uniform designs, and even to both interior and boundary points. It is shown that the local linear regression smoothers have high asymptotic efficiency (i.e., can be 100% with a suitable choice of kernel and bandwidth) among all possible linear smoothers, including those produced by kernel, orthogonal series, and spline methods. The finite sample property of the local linear regression smoother is illustrated via simulation studies. Nonparametric regression is frequently used to explore the association between covariates and responses. There are many versions of kernel regression smoothers. Some estimators are not good for random designs, such as in observational studies, and others are not good for nonequispaced designs. Furthermore, most nonparametric regression smoothers have “boundary effects” and require modifications at boundary points. However, the local linear regression smoothers do not share these disadvantages. They adapt to almost all regression settings and do not require any modifications even at boundary. Besides, this method has higher efficiency than other traditional nonparametric regression methods.