The Berry–Esseen bound for $\rho$-mixing random variables and its applications in nonparametric regression model

In this paper, the Berry–Esseen bound for $\rho$-mixing random variables with the rate of normal approximation $O(n^{-1/6}\log n)$ is established under some suitable conditions. By using the Berry–Esseen bound, we further investigate the Berry–Esseen bound of sample quantiles for $\rho$-mixing random variables. The rate of normal approximation is shown to be $O(n^{-1/6}\log n)$ under some suitable conditions. In addition, the asymptotic normality of the linear weighted estimator for the nonparametric regression model based on $\rho$-mixing errors is studied by using the Berry–Esseen bound that we established. Some new results are obtained in the paper under much weaker dependent structures.