| Summary: | 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 < ∞ $E|Y_{1}|^{s}<\infty$ ( s > 2 $s>2$ ), the convergence rate | m ˆ n ( x ) − m ( x ) | = O P ( a n ) + O ( h 2 ) $|\widehat{m}_{n}(x)-m(x)|=O_{P}(a_{n})+O(h^{2})$ and uniform convergence rate sup x ∈ S f ′ | m ˆ n ( x ) − m ( x ) | = O p ( a n ) + O ( h 2 ) $\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-m(x)|=O_{p}(a_{n})+O(h^{2})$ are presented, if a n = ln n n h d → 0 $a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\rightarrow0$ . We generalize some results in Shen and Xie (Stat. Probab. Lett. 83:1915-1925, 2013).
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