On complete consistency for the weighted estimator of nonparametric regression models

In this paper, we consider the following nonparametric regression model: where xni are known fixed design points from A, where A< subset of>Rd is a given compact set for some d1, f() is an unknown regression function defined on A and epsilon ni are random errors, which are assumed to widely orthant dependent (WOD, for short). Firstly, a general result on complete convergence for partial sums of WOD random variables is obtained, which has some interest itself. Based on some mild conditions and the complete convergence result that we established, we further establish the complete consistency of the weighted estimator in the nonparametric regression model, which improves the corresponding one of Wang et al. (TEST 20:607-629, 2014). As an application, the complete consistency of the nearest neighbor estimator is obtained. Finally we provide a numerical simulation to verify the validity of our result.