Design-adaptive Nonparametric Regression

Abstract

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.

Key Words:

• Boundary effects
• Kernel estimator
• Linear smoother
• Local linear regression
• Minimax efficiency