Asymptotic normality of DHD estimators in a partially linear model

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