The Berry–Esseen bounds of the weighted estimator in a nonparametric regression model

Consider the following nonparametric model: $$Y_{ni}=g(x_{ni})+ \varepsilon _{ni},1\le i\le n,$$ Y ni = g ( x ni ) + ε ni , 1 ≤ i ≤ n , where $$x_{ni}\in {\mathbb {A}}$$ x ni ∈ A are the nonrandom design points and $${\mathbb {A}}$$ A is a compact set of $${\mathbb {R}}^{m}$$ R m for some $$m\ge 1$$ m ≥ 1 , $$g(\cdot )$$ g ( · ) is a real valued function defined on $${\mathbb {A}}$$ A , and $$\varepsilon _{n1},\ldots ,\varepsilon _{nn}$$ ε n 1 , … , ε nn are $$\rho ^{-}$$ ρ - -mixing random errors with zero mean and finite variance. We obtain the Berry–Esseen bounds of the weighted estimator of $$g(\cdot )$$ g ( · ) . The rate can achieve nearly $$O(n^{-1/4})$$ O ( n - 1 / 4 ) when the moment condition is appropriate. Moreover, we carry out some simulations to verify the validity of our results.