Bahadur representations of M-estimators and their applications in general linear models

Abstract

Consider the linear regression model $y_i=x_i^T\beta +e_i,i=1,2,\dots ,n,$ where $e_i=g(\dots ,{\epsilon }_{i-1},{\epsilon }_i)$ are general dependence errors. The Bahadur representations of M-estimators of the parameter β are given, by which asymptotically the theory of M-estimation in linear regression models is unified. As applications, the normal distributions and the rates of strong convergence are investigated, while $\{{\epsilon }_i,i\in Z\}$ are m-dependent, and the martingale difference and $(\epsilon ,\psi )$ -weakly dependent.

Keywords: Bahadur representation; Linear regression models; M-estimate; Normal distribution; Rate of strong convergence.