Strong representations for LAD estimators in linear models

Consider the standard linear model y_i=z_iβ+e_i, i=1, 2,..., n, where z_i denotes the ith row of an n x p design matrix, β∈ℝ^p is an unknown parameter to be estimated and e_i are independent random variables with a common distribution function F. The least absolute deviation (LAD) estimate {Mathematical expression} of β is defined as any solution of the minimization problem {Mathematical expression} In this paper Bahadur type representations are obtained for {Mathematical expression} under very mild conditions on F near zero and on z_i, i=1,..., n. These results are extended to the case, when {e_i} is a mixing sequence. In particular the results are applicable when the residuals e_i form a simple autoregressive process.