The asymptotic normality of the linear weighted estimator in nonparametric regression models

Consider the following nonparametric regression model: Yni=g(xni)+εni,i=1,2,…,n,n≥1, $$ Y_{ni} = g(x_{ni}) + \varepsilon _{ni},\quad i = 1, 2, \ldots, n,n\ge 1, $$ where xni are known fixed design points from A⊂Rd$A\subset \mathbb {R}^d$ for some positive integer d ⩾ 1, g( · ) is an unknown regression function defined on A and ϵni are random errors. Under some suitable conditions, the asymptotic normality of the linear weighted estimator of g in the nonparametric regression model based on ρ-mixing errors is established. The key techniques used in the paper are the Rosenthal type inequality and the Bernstein’s bigblock and small-block procedure. The result obtained in the paper generalizes the corresponding ones for some dependent sequences.