Asymptotic normality of DHD estimators in a partially linear model

Abstract

The paper studies a partially linear regression model given by $$\begin{aligned} y_i=x_i^T\beta +f(t_i)+\varepsilon _i,i=1,2,\ldots ,n, \end{aligned}$$ where \(\{\varepsilon _i,i=1,2,\ldots , n\}\) are independent and identically distributed random errors with zero mean and finite variance \(\sigma ^2>0\). Using a difference based and the Huber–Dutter (DHD) approaches, the estimators of unknown parametric component \(\beta \) and root variance \(\sigma \) are given, and then the estimation of nonparametric component \(f(\cdot )\) is given by the wavelet method. The asymptotic normality of the DHD estimators of \(\beta \) and \(\sigma \) are investigated, and the weak convergence rate of the estimator of \(f(\cdot )\) is also investigated. In addition, for stationary \(m\)-dependent sequence of random variables, the central limit theorem is also obtained. At last, two examples are presented to illustrate the proposed method.