Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency

Abstract

In this paper, for a wide class of nonparametric regression models, new local linear kernel estimators are proposed that are uniformly consistent under close-to-minimal and visual conditions on design points. These estimators are universal in the sense that their designs can be either fixed and not necessarily satisfying the traditional regularity conditions, or random, while not necessarily consisting of independent or weakly dependent random variables. With regard to the design elements, only dense filling of the regression function domain with the design points without any specification of their correlation is assumed. This study extends the dense data methodology and main results of the authors’ previous work for the case of regression functions of several variables.

Keywords

• nonparametric regression
• local linear estimator
• uniform consistency
• fixed design
• random design
• strongly dependent design elements