Uniform convergence of estimator for nonparametric regression with dependent data

Abstract

In this paper, the authors investigate the internal estimator of nonparametric regression with dependent data such as α-mixing. Under suitable conditions such as the arithmetically α-mixing and $E|Y_{1}|^{s}<\infty$ ( $s>2$ ), the convergence rate $|\widehat{m}_{n}(x)-m(x)|=O_{P}(a_{n})+O(h^{2})$ and uniform convergence rate $\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-m(x)|=O_{p}(a_{n})+O(h^{2})$ are presented, if $a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\rightarrow0$ . We generalize some results in Shen and Xie (Stat. Probab. Lett. 83:1915-1925, 2013).